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COPVRIGHT DEPOSnV 



ELEMENTS 



OF 



DIFFERENTIAL CALCULUS. 



BY 

EDGAR W. BASS, 

Colonel United States Artny, Retired. Professor of Mathematics in the 
U. S. Military Academy April 17, 1878, to October 7, 1898. 



THIRD EDITION. 
FIRST THOUSAND. 



NEW YORK : 

JOHN WILEY & SONS. 

London: CHAPMAN & HALL, Limited. 

1905 



LIBRARY of CONGRESSJ 


Two Copies 


Receivea 


DEC 21 


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1 COPY 


XXC NO! 


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1904, 



Copyright, 190 1 
. BY 

EDGAR W. BASS. 



ROBERT DRUMMOND, ELBCTROTYF2R AND PRINTER, NEW YORK. 



PREFACE. 



This text-book has been prepared for the use of the 
Cadets of the U. S. Military Academy who begin the sub- 
ject with a knowledge of the elements of Algebra, Geometry, 
and Trigonometry which ranges from fair to excellent. 
The time allotted to the subject (ten and one half weeks), 
and the requirements of the subsequent courses, especially 
Mechanics, Ordnance and Gunnery, and Engineering, limit 
and determine the scope of the work. 

My experience leads me to the belief that the more rig- 
orous and comprehensive method of infinitesimals is suit- 
able only for a treatise, and not for a text-book intended 
for beginners. 

At the same time I believe that any presentation of the 
subject, no matter how elementary, should in no manner 
prejudice the student against any established method. On 
the contrary, it should, I think, endeavor to lead him to an 
understanding of the relations between those in general use, 
and, above all, it should aim to construct the best possible 
ground work for the subsequent study of the subject treated 
in the most rigorous and extended form. 

The principle of interchange of infinitesimals, which con- 
sists in replacing one infinitesimal by another when unity 



IV PREFA CE. 

is the limit of their ratio, has been used to overcome the 
difficulty encountered by beginners in the determination 
of the limits of ratios of infinitesimals. 

To the Officers of the U. S. Army who have taught the 
subject with me, and in many cases to my pupils, I am 
greatly indebted for much valuable assistance. 

To Captain Wm. Crozier, Lieut. J. A. Lundeen, Lieut. H. 
H. Ludlow, and Lieut. F. Mclntyre I am under obligations 
for many demonstrations and solutions. 

Associate Professor W. P. Edgerton has been my collab- 
orator throughout the work, and lo him much credit is due 
for numerous demonstrations, improvements, and sugges- 
tions. 

I have added a list of the works of authors which I have 
freely consulted in the preparation of this book, for the 
purpose of acknowledging my indebtedness to them, and 
for the benefit of students who may desire to extend their 
knowledge of the subject. 



Edgar W. Bass. 



West Point, N. Y., June 15, li 



PREFACE TO THE SECOND. EDITION, 



For the corrections and changes made in this edition I 
am indebted to the Department of Mathematics, U. S. M. A. 
My thanks are especially due to Professor W. P. Edgerton, 
Associate Professor Chas. P. Echols, Lieut. George B. 
Blakely, and Lieut. F. W. Coe. 

Edgar W. Bass« 

524 Fifth Avenue, New York City, 
Tune I, iQOi. 



PREFACE TO THE THIRD EDITION. 



I AM indebted to Professor Chas. P. Echols for the cor- 
rections and changes made in this edition. 

Edgar W. Bass. 
Bar Harbor, Maine, 

August 28, 1904. V 



LIST OF AUTHORS WHOSE WORKS HAVE BEEN 
CONSULTED IN THE PREPARATION OF THIS 
BOOK. 





AMERICAN. 


Church, 


Newcomb, 


Rice and Johnson, 


Bowser, 


Byerly, 


Hardy, 


Taylor, 


Osborne. 




ENGLISH. 


Todhunter, 


Greenhill,* 


Williamson, 


Price,* 


Edwards,* 


Haddon, Examples, 




FRENCH. 


Bertrand,* 


Houel,* 


Jordan,* 


Haag,* 


Duhamel,* 


Serret.* 



Harnack's Introduction to the Calculus, translated from 
the German by Cathcart. is a rigorous treatment of the 
subject. 

* Treatises 



CONTENTS. 



DIFFERENTIAL CALCULUS. 
INTRODUCTION. 

DEFINITIONS. NOTATION AND FUNDAMENTAL PRINCIPLES. 

PAGE 

Chapter \. Constants, Variables, and Functions ... i 

II, Principles of Limits 25 

III. Rate of Change of a Function 44 

PART I. 

DIFFERENTIALS AND DIFFERENTIATION. 

IV. The Differential and Differential Coef- 
ficient 59 

V. Differentiation of Functions 72 

VI. Successive Differentiation... 121 

VII. Implicit Functions and Differential Equa- 
tions 141 

VIII. Change OF the Independent Variable 154 

PART II. 

ANALYTIC APPLICA TIONS. 

IX. Limits of Functions which assume Indeter- 
minate Forms 158 

X. Developments 170 

XL Maximum and Minimum States 206 

ix 



X CONTENTS. 

PART III. 

GEOMETRIC APPLICA TIONS. 

PAGE 

Chapter XII. Tangents and Normals 243 

XIII. Asymptotes „.. 256 

XIV. Direction of Curvature. Singular Points.. 271 
XV. Curvature of Curves 289 

XVI. Involutes and Evolutes 304 

XVII. Orders of Contact of Curves and Oscu- 
lating Lines 317 

XVIII. Envelopes 324 

XIX. Curve-tracing 335 

XX. Applications to Surfaces 348 



DIFFERENTIAL CALCULUS 



INTRODUCTION. 

DEFINITIONS, NOTATION AND FUNDAMENTAL 
PRINCIPLES, 



CHAPTER I. 
CONSTANTS, VARIABLES AND FUNCTIONS. 

I. In the Calculus quantities are divided into two 
general classes, constants and variables. 

A Constant is a quantity that has, or is supposed to 
have, an absolute or relative fixed value. 

A Variable is a quantity that is, or is supposed to be, 
continually changing in value. 

In general, constants are represented by the first letters 
of the alphabet, and variables by the last ; but they should 
not, therefore, be confused with the known and unknown 
quantities of Algebra. 

The same quantity may sometimes be either a variable 
or a constant, depending upon the circumstances under 
which it is considered. Thus, in the equation of a curve, 
the coordinates of its points are variables ; but in the 



,2 DIFFERENTIAL CALCULUS. 

equation of a tangent to the curve, the coordinates of the 
point of tangency are generally treated as constants. It is, 
therefore, necessary to determine from the circumstances, 
or object in view, which quantities are to be regarded as 
variables, and which as constants, in each discussion. 

In general, any or all of the quantities represented by 
letters in any mathematical expression or equation may 
have definite values assigned to them, and be regarded as 
constants ; or they maybe considered as changing in value, 
and treated as variables. Thus, in the expression ^7Tr\ r is 
a constant if we suppose it to represent the radius of a par- 
ticular sphere ; but if r is considered as changing in value, 
it will be a variable. In the first case, A^nr^ is a constant, 
and measures the surface of a particular sphere ; but when 
r is variable, ^nr" is also variable, and represents the surface 
of any sphere no matter how much it may increase or 
diminish. 

It should not be understood, however, that we may in 
all cases treat quantities as constants or variables at pleas- 
ure without affecting the character of the magnitude rep- 
resented by the expression or equation. For example, n 
is generally assumed to represent the ratio of the circum- 
ference of any circle to its diameter, which ratio is invariable. 
If a different value be assigned to jt, the expression ^nr^ 
will not measure the surface of a sphere whose radius is r. 

In some cases variation in a quantity changes the di- 
mensions of the magnitude represented by the expression 
or equation ; in others it changes the position only ; and 
again it may change the character of the magnitude. Thus, 
if we suppose R to vary in the equation 

(oc - ay + (j, - /?)' = ^, 



CONSTANTS, VARIABLES AND FUNCTIONS. 3 

we shall have a series of circles differing in size ; but by 
changing a or (3 and not R the position only will be 
affected. 

By changing b'^ within positive limits, the equation 
«y + b'^^'^ = ^^b"^ represents different ellipses, but negative 
values for b"^ cause the equation to represent hyperbolas. In 
general, however, constants are supposed to have fixed 
values in the same expression, unless for a particular dis- 
cussion it is otherwise stated. 

FUNCTIONS. 

2.. A quantity is a function of another quantity when 
its value depends upon that of the second quantity. Thus, 
\ax is a function of 4, <2, and x. In general, any mathemat- 
ical expression which contains a quantity is a function of 
that quantity. If, however, a quantity disappears from an 
expression by reduction or simplification the expression is 
not a function of that quantity. Thus, oc" -|- [c ^ x){c — x), 
ax/bx^ and tan x cot x^ are not functions of x. 

3* A function of a single variable is one whose value 
depends upon that of a single variable and varies with it. 
Thus, 



4^y(i — jic"), yr^x^ + 2/jc, log (a ~\- jc), sec x^ 

in which x is the only variable, are functions of a single 
variable. 

Any function of a single variable is also a variable, and 
varies simultaneously with the variable. 

4. The relation between a function of a variable and its 
variable is one of mutual dependence. Any change in the 
value of one causes a dependent variation in that of tlie 
other. Either may, therefore, be regarded as a function of 



4 DIFFERENTIAL CALCULUS. 

the other ; and they are called inverse functions. Thus, if 
X passes from the value 2 to 3, the function 20^ will vary 
from 8 to 18 ; and conversely, x will increase from 2 to 3, if 
2X^ changes from 8 to 18. \i x be again increased the same 
amount, that is from 3 to 4, the function will increase from 
18 to 32. Similarly, with other functions we shall find that, 
in general, equal changes in the variables do not give equal 
changes in the corresponding functions. 

It is therefore necessary, in referring to a change in a 
function correspondiiig to a change in the variable, to con- 
sider the states from which and to which the function and 
variable change, as well as the amount of change in each. 
With that understanding, corresponding changes in a func- 
tion and its variable are mutually dependent. 

In the equation of a curve^ the ordinate of any point is a 
function of the abscissa, and the abscissa is the inverse 
function of the ordinate. 

The function is considered as dependent, and the vari- 
able as independent ; for which reason, the latter is called 
the independent variable^ or simply the variable. 

Representing a function of x^ as ^^byjj;, we havejv = x^\ 
solving with respect to x^ we have x^=^ yy\ a form express- 
ing directly Jt: as a function of j'. 

The difference in form in the following important exam- 
ples of direct and inverse functions should be observed. 

Having, _y = x^\ then x = \/y. 
" y — a-\- x\ " X = y — a. 
" y = ax; " X = y/a. 



CONSTANTS, VARIABLES AND FUNCTIONS. 5 

5. A state of a function corresponding to a value or ex- 
pression for the variable is a result obtained by substituting 
the value or expression for the variable in the function. 
Thus, 

— 00 , — i6rt;, — 2^, o, 2^, 16^. 00, 

are the states of the function 2ax^ corresponding, respec- 
tively, to the values or expressions for x^ 

— 00, —2, —I, o, I, 2, 00, 

and 

o, 1/2, S/TfV, I, o, —I, o, 

are the states of the function sin corresponding, respec- 
tively, to the expressions or values of 0, 

o, ^/6, 7r/4, 7r/2, ;r, 37r/2, 27t. 

A function of a variable has an unlimited number of 
states. It may have equal states corresponding to different 
values of the variable ; and it may have two or more states 
corresponding to the same value of the variable. Thus, 

5 and I, 7±l/i2, 13 and 5, i3±V24, 25 and 13, 

are the states of the function 2X -\- \±. V4X, corresponding, 
respectively, to the values of x, 

I, 3» 4, 6, 9. 

Trigonometric functions have equal states for all angles 
differing by any entire multiple of 27t. 

In connection with any state of a function corresponding 
to any value of the variable, it is frequently necessary to 
consider another state of the function, which results from 



6 DIFFERENTIAL CALCULUS. 

increasing the value of the variable corresponding to the 
first state by some convenient arbitrary amount. In order 
to distinguish between these two states of the function, the 
first is designated as a primitive state, and the other as its 
new or second state. 

Any arbitrary amount by which the variable is increased 
from any assumed value is called an increment of the vari- 
able. It is generally represented by the letter h, or k, or by 
A written before the variable ; as, Ajc, read " increment of 
^"* 

Let x' represent any particular value of x, and h, or t\x' , 
its increment ; then will 2ax''^ and 2a{x' + Hf., or 
2a(x' + A x'Y, represent, respectively, the primitive and 
new states of the function 2ax^, corresponding to x' and its 
increment h, or l\x'. The general expression 2a{x -\- Ji)'^ 
represents the second state of any primitive state of the 
function 2ax^, and from it we obtain the second state corre- 
sponding to any particular primitive state by substituting 
the proper value of x. 

6. A function of two or more variables is one which de- 
pends upon two or more variables and varies with each. 
Thus, 

X sin y, xy, x^, y log x, x^ -j" y xy — 3_);, 

are functions of x and_>' ; and 

^■\- y -\- ^y y + tan x/z, z ^vciipc^y), \^x^ -\- y^ + log -2^, 

are functions of x, y and z. Each variable is independent 
of the others. Particular values or expressions may be 

* Increment as here used is in an algebraic sense, and includes a 
decrement, which is a negative increment. In general, an assumed 
increment of a variable is regarded as positive. 



CONSTANTS, VARIABLES AND FUNCTIONS. 7 

assigned to one or more of the variables, and the result dis- 
cussed as a function of the remaining variables. A func- 
tion of two or more variables possesses all of its properties 
as a function of each variable. By substituting in the 
function 2x'' -\-y, any assumed value ior y, as 5, the result 
2X + 5 is a function of a single variable. 

7. A quantity is a function of the sum of two variables 
when every operation indicated upon either variable includes 
the sum of the two. Thus, 



3^ ^ X ±y, sin {x ± y), log {x ±y), a^^y, 

and all algebraic expressions which may be written in the 
form 

A{x±yY-\-B{x ±yY-^ -\- -f H, 

in which A^ By etc., are constants, are functions of the sum 
of the two variables x and ±y. 

Mx(x-{-yY, Vx —y — 2y, Vx-]-y, ^+^, xsin{x—y)y 

are not functions of the sum of x and j. 



sin(^-^±/), A{x'±yr, 3^og{x^±/), V2{x'±/)-{-ja, 

are functions of the sum of the two variables x'' and ±y% 
but not of the sum of x and ±y. 

2{dVx-{- ay""), cos'(<^V^ + ^/), 2 A/log {bVx-^ ^/ — 3^), 

are functions of the sum of the two variables d V^and ay"^. 

In any function of the sum of two variables, a single 

variable may be substituted for the sum, and the original 

function expressed as a function of the new variable. 



O DIFFERENTIAL CALCULUS. 

Thus, z may be substituted for [x -^ y) in the function 
(^{.^ + j)") giving the function in the form az^. In a similar 
manner we may write 

tan {x — y) = tan z, a^+^ = a^^ 2aV\og{x—y) = 2a\^\og z; 

but it must be remembered that z in the new form is a 
function of the two variables x and y. 

8. A state of a function of two or more variables, corre- 
sponding to a set of values or expressions for the variables, 
is the result obtained by substituting those values or ex- 
pressions for the corresponding variables. Thus, 



— 20, — 6 



are states of the function ^x -\- T^y -\- 2 corresponding, re- 
spectively, to the values or expressions for x andji^, 

(-4,-2), (-2,0), (-8, + 10), (0,1), (2,5); 

and 

o, 4/1/3, I, ^3y c», 

are states of the function tan {x -\- y) corresponding, re- 
spectively to the values or expressions for x andji^, 

(0,0), {7r/lZ,7t/g), {7t/l2,7r/6), {27t/g,7t/^), (o, TT^). 

Any function, in which all of the variables are indepen- 
dent, is a variable, and has an unlimited number of states. 

9. A function of several variables may be equal to some 
constant value or expression ; in which case one of the 
variables is dependent upon the others. Thus, the first 
member of the equation 2x -\- 3;' = 7 is a function of the 
two variables, x and 7 ; but x and j' are mutually dependent. 



CONSTANTS, VARIABLES AND FUNCTIONS. Q 

Any equation containing n variables expresses a depen- 
dence of each variable upon the others; and there are only 
n — \ independent variables in such an equation. In other 
words, the number of independent variables in any equation 
is one less than the total number of variables. 

In any group of equations, the number of independent 
variables is equal to the total number of variables less the 
number of independent equations. 

10. An Algebraic function is one that can be expressed 
definitely by the ordinary operations of Algebra ; that is, by 
addition, subtraction, multiplication, division, formation of 
powers with constant commensurable exponents, and ex- 
traction of roots with constant commensurable indices. 

Algebraic functions have particular names based upon 
peculiarities of form. 

A ratioiial function of a variable is one in which the vari- 
able is not affected by a fractional exponent. 

An integral function of a variable is one in which the 
variable does not enter the denominator of a fraction, or in 
other words, is not affected by a negative exponent. 

^m _|_ ^jt;;« - I _j_ ^j^;;« - 2 _j_ Gx -{- H, 

in which m is a positive integer, and A, B, etc., do not con- 
tain X, is a rational and integral function of x. The coeffi- 
cients A^ B, etc., may be irrational or fractional. 

A rational integral function of a variable is also called an 
entire function of that variable. 

A linear function of two or more variables is one in 
which each term is of the first degree with respect to the 
variables. 

Thus, 2X -f 3j -f 72; is a linear function of x^ y and z. 

A function is homogeneous with respect to its variables 



lO DIFFERENTIAL CALCULUS. 

when all of its terms are of the same degree with respect to 
them. 

A linear function is a homogeneous function of the first 
degree. 

II. A Transcendental function is one that cannot be 
expressed definitely by the ordinary operations of Algebra. 

In general, a transcendental function may be expressed 
algebraically by an infinite series. 

Transcendental functions include expo7iential^ logarithmicy 
trigonometric^ inverse trigonometric^ hyperbolic and inverse 
hyperbolic functions. 

An Exponential function is one with a variable, or an in- 
commensurable constant, exponent; as, 

A Logarithmic function is one that contains a logarithm 
of a variable; as, 

log X, log {a-\-y)^ 2ax' — ::i;/log x.* 

A Trigonometric function is one that involves the sine, 
or cosine, or tangent, etc., of a variable angle ; as, 

cot X, sec 2X^, (^— sin x)/x^^ versin' x. 

An Inverse Trigonometric function is one that contains 
an angle regarded as a function of a variable sine, or cosine, 
or tangent, etc. 

Sin~^jj;, tan""'j>/, read " the angle whose sine is j " ; " whose 
tangent isj'" ; are symbols used to denote such functions. 
Having given y = versin x, then x ■= versin'^j/ ; and if 
u = cos y^ then y = cos~^u, etc. 

^ Napierian logarithms will always be considered unless some other 
base, as a, is indicated by loga. 



CONSTANTS, VARIABLES AND FUNCTIONS. I I 

12. Hyperbolic Functions.— From Trigonometry, we 

have 

X' ^ x' ^' 

sm ^ = ^ r h -T , f- etc.; 

13. li ll 

Uv *A/ i/v 

cos X = 1 -f- — — r— 4" etc. 

2 4 6 



Placing V — I = z, and substituting xi for x^ we have 

x^ x^ oc' 
sin xi = i{x^— -^ — - -f -— + etc.); 

|3 li 11 

COS ^z =: I -| [- [- -— -f etc. 

2 |4 |6 

From Algebra, we have 

x^ x^ X* 

e" =1 + ^-1 ^ _|. + etc.; 

^ 13 li 
^"^= \ — X -\- r [- -T etc. 

^ ii l± 

Hence, 

^^ + ^-^ , Jt:' Jt:' ^' 

— T-=' + T+-l^+|^+^*<=•' 
Therefore, 

sin xi = i(e'' — ^~^)/2; cos-^/=: (^^ + e''^)/2. 

cos ^7 is real, and is called the hyperbolic cosine of x. It 
is generally written cosh x. The real factor in the sine of 
xi is called the hyperbolic sine of x, and is written sinh x. 
Thus, 

sinh X— {e'' — ^~*)/2 ; cosh x— (^^ + ^"'')/2. 



12 DIFFERENTIAL CALCULUS. 

From which, 

cosh'^ X — <SA.X\\v ^ = I. 

Comparing with x^ — y^ — i, we see that cosh x and sinh 
X may be represented by the coordinates of points of an 
equilateral hyperbola, referred to rectangular coordinate 
axes, with the original its centre. Hence the name hyper- 
bolic functions. The functions sinh x and cosh x are not 
periodic functions for real values of x^ but increase with x 
indefinitely. 

By analogy with trigonometric functions, other hyper- 
bolic functions are defined, and written, as 

sinh X 

tanh X = — - — 

cosh J^: 

coth X = — - — 
tanh X 

sech X — 

cosh X e"^ -\- e~'^* 

cosech X — -r-, — = — ^. 

Sinn X e^ — e~ 

It follows that 

tanh'' X + sech^ x — \ — coth'' x — cosech'' x 

13. Inverse Hyperbolic Functions — Writing J^^ = sinh x, 

we have x = sinh"^ y. 



e^ 


— e 




e"" 


-\-e- 


x> 


e^ 


+ e- 


-X 


e^ 


— e~ 


x'> 




2 





From 7 = sinh x — (^^ — e '*')/2, we find e"^ = y±Vi +^1 
hence, sinh"^ y z=i x= log (j±Vy^ + i)* Similarly, 
y = cosh X = (e"" -\- <?~^)/2 gives cosh~^ y=\og(j ±V/ — 1); 

c^ — c~^ i-l-y 
y = tanh x — -— gives tanh"' y -- I log ; 



CONSTANTS, VARIABLES AND FUNCTIONS. 



coth X = ~ — gives coth ^ y ■= \ log 



e — e y — I 
y = sech X = ^ ^ ^_^ gives sech-^j/ == log ^ ; 



r=cosecn:r=:— — gives cosech y = log -^-^. 

,?'*' — ^^ ;; 

14. Explicit and Implicit Functions. — When a function 
is expressed directly in terms of its variable or variables, it 
is an explicit function ; otherwise it is an implicit function. 

Thus, in the equations 

J = 2^' + 3^, y = tan'x, y = f, y = log 2ax\ y = f{x, z), 

y is an explicit function of the variables in the second 
members, and in the equations 

^y + 3V=:^'3^ f^\ogx\ f = r'-x\ f{y,x)=o, 

y is an implicit function of x. 

The relation between an implicit function and its vari- 
ables is sometimes expressed by two equations. Thus, 
y = 32/j u^ — Vx, in which y is an implicit function of x. 

y =/W> u= (p{x)', and y — f{u), x = (p{u), 

are forms expressing jf as an implicit function of x. 

15. Increasing and Decreasing Functions — A function 
that increases when a variable increases, and decreases 
when that variable decreases, is an increasing function of 
that variable. Thus, 2X^ y.-r^, 2^, ax^/b, tan x, are increas- 
ing functions of x. 

A function that decreases when a variable increases, and 
increases when that variable decreases, is a decreasing 



14 DIFFERENTIAL CALCULUS. 

function of that variable. Thus, i/.r, {c — xf^ b/ax^^ cot.r, 
are decreasing functions of x. 

Functions are sometimes increasing for certain values of 
the variable, and decreasing for others. Thus, {c — a)' is 
an increasing function for all values of x greater than c\ 
but decreasing for all values of x less than c. 2ax'^ is an 
increasing function when x is positive, and decreasing when 
X is negative. The positive value of j = ± S^ r' — x^ is an 
increasing function for values of x from — r to o, but de- 
creasing for values of x from o to + r. The negative 
value of jv is a decreasing function for negative values of x^ 
and increasing for positive values of x. sin x^ cos x^ sec x^ 
vers X, are increasing functions for some values of the vari- 
able and decreasing for others. 

i6. Continuous and Discontinuous Functions. — A func- 
tion is continuous between states corresponding to any two 
values of a variable when it has a real state for every inter- 
mediate value of the variable, and as the difference between 
any two intermediate values of the variable approaclies 
zero, the difference between the corresponding slates also 
approaches zero. Otherwise a function is discontinuous 
between the states considered. 

The varying height of a growing plant is a continuous 
function of time. 

If for any value of the variable a function is unlimited, 
imaginary or changes from one state to another without 
passins: through all intermediate states, the continuity of 
the function is broken at the corresponding state. 

±.Sf 2px is continuous between states corresponding to 
jc = o and ■%■ = oo . 



-^b/a y a^ — x"^ is continuous between states correFpond- 
ing to X = —a and x ^= -\- a. 



CONSTANTS, VARIABLES AND FUNCTIONS. 1 5 



±.bla\ x' — a' is continuous between states correspond- 
ing to 

^ = — 00 and jc = — ^, ^ = (2 and ji; = oo , 

but is discontinuous between states corresponding to 
X =^ — a and x ■= a. 

Sin X, cos :r, e"", and all entire algebraic functions are 
always continuous. 

A continuous function in passing from any assumed state 
to another must pass through all states intermediate to 
those assumed ; but it may have intermediate states greater 
or less than the states assumed. Thus, the function 



^/r'^ — x^ is continuous between the states o and r/2 V^, 
which correspond to x = — r and x = r/2 ; but it is greater 
when X ^^ o than either of the states considered. 

A function always continuous changes its sign only by 
passing through zero ; but a discontinuous function may 
change its sign without passing through zero. 

Unless otherwise stated, functions will be regarded as 
continuous in the vicinity of states under consideration. 

17. Functional Notation. — A function of any quantity, 
as .T, is generally represented thus, /(x), read "function of 
x" Other forms are also used; as, /'{^), ■F{x), F^x), 
0Gt), 0'W, 4-{x), il:^(x). 

Thus, ax/(i-^x) maybe represented by /(.t). The/, 
or exterior symbol, is called the functional symbol, or sym- 
bol of operation. It represents the operations involved in 
any particular function. Thus, having f{x) = ax/{i -\- x), 
f indicates that x is to be multiplied by ^, and that the 
product is to be divided by i -f- x. Its significance re- 
mains unchanged throughout the same discussion or subject, 
and placed before the parenthesis enclosing any other quan- 



1 6 DIFFERENTIAL CALCULUS. 

tity it indicates that the quantity enclosed is to be sub- 
jected to the same operations that x is in the expression 
axl[\ -j- x). Thus, 

. /(^) = r^., /(sin 0) = "^ ^'" "^ 



1+2 I -|- Sin 

In order to represent different functions of the same 
quantity the functional symbol only is changed. Thus, if 
F{x) is selected to represent 2 \^bx, then some other forn^ 
as FX^), or 0(^), etc., should be taken to denote ^cx^ + 2X 

Different functions of different quantities are represented 
by different symbols within and without the parentheses. 
Thus, Vx'' — a^ and 4>'V(i — y^) may be denoted by f{pc) 
and <p{y), respectively. 

A function of x^ is written fix'') or F{x^), etc., and the 
square of a function of x is designated by/(:^) or <p{x) , 
etc. 

When the quantity is represented by a simple symbol, 
the notation fx, (px, Fx'^, tj^x^^ etc., is frequently used. 

T^cVmy'^ may be expressed as a function of my' by some 
form, 2L'S, f{my'^) ox f'{iny'\ etc. 

Having represented az' by /(2), and 3^ \^az' by F{az''), 
we may write y Vaz'' = F{az^) = F[/{z)]. 

Having a"" = (p{^), and b Va-^ = ^'(^*), we write 

2db ya^ -\- h ^ ' 

An expression containing several different functions of a 
variable, as 2ax:' — log x -j- 3 sin jjc, may be considered as 



CONSTANTS, VARIABLES AND FUNCTIONS. 1/ 

a function of the several functions of the variable, and 
represented thus: 

4>V^{^y)^ ^Wj/(-^)] represents a function of three differ- 
ent functions of different variables. 

P'unctions of two variables are denoted thus: /(-^j >'), 
f\x,y), F{y, z), (p{x, y), tp{x, z), ip^{x, z), etc.; and func- 
tions of three variables by F{x,y, z), ^p{r, s, t), etc. 

Functions of any number of variables are indicated simi- 
larly by writing all the variables within the parentheses. 

In all cases a functional symbol indicates the same oper- 
ations in any one subject. 

Thus, if f{x, y) = ax -{- by, then f{s, t) = as -\- bt\ /{i, 3) 
= 2a -\- ^b] /(o, m) = bm. 

Having 0(^, z,y) = 2x — cz-^-y"^, then (p{r, s, t) = 
2r-cs-\- t\ 

Functions are frequently represented by single letters; 
thus, ± y Bi^ — x^ may be represented by y, giving 
_y = ± y/ R^ — x^\ and/(^, y) by z, giving z = f(x,y). 

F{x -\-y)y f{x + Ji), ^{/ + r^) are forms denoting func- 
tions of the sum of two variables. §7. 

ILLUSTRATIONS. 

Having /(^) -x^-\- Px^-' + Qx^-^ + . . .-\-U,m which F, Q, 
etc., do not contain x, then 

f{3bc) = {3bcy^ + F{3bc)^-^ + . . . + ^7. 

f{a - x) = {a- xy^ -I- P{a - x)»'-^ + . . . -j- i/ 

f{o) = o^-\- Fo»'-^ + .:..-{- W 

/(.')=0 + i'(.^)— + .. .. + 1/. 



1 8 DIFFERENTIAL CALCULUS. 

Having then 

(P{a) = Ao' + ca, <p{x + r) = 4(x -^ yf + c{x -\-y). 

ip{ax^) = 4{ax^)'^ -f c{ax^), ^(sin G) ^ 4 sin^S + ^ sin 6. 

F(x) = a^ , ^{x -\- y) = a^^y = a^ y^ay = F{x) X F{v). 

F{x)) z= a'^y, F(x)^ = Fjf = (a^)^ ={ay = F(xy). 

/x = \ogx, /(limit ^) = log (limit ;i;). 

fnx=a^, fn (limit x) = a"«»" ^. 

n y- c\ w — {jvy 

If (p(i) = \z, i>{x) = sax, and J^(w) = ^^ _ ^j, > 

then ^(^[^(.)]) = '^^''^^~^-(s''Viy ^^^^^^ 



If 0(jf, jf) — 2j; + sin_j/, and i}){z) = 2>V !^, then ?^f0(jr, jj/)] = 3y2^-|- sin 
If /(-^j J' ^) = 7<^^r^/2, and /"(j)^) = \^y 



then 



-^(^[/(-O'.-)])] = 2a^<'-V^)'. 



18. Lines are classed as algebraic or transcendental 
according as their equations involve algebraic functions 
only or contain transcendental functions. 

Any portion of any line may be considered as generated 
by the continuous motion of a point. The law of its motion 
determines the nature and class of the line generated. 

Let s represent the length of a varying portion of any 
line in the coordinate plane XY, of which the equation in 
X and y is given. .9 depends upon the coordinates of its 
variable extremities, and varies with each; but the equation 
of the line establishes a dependence between these coordi- 
nates. Hence, j" is a function of one independent variable 
only. 

If the line is in space, its two equations establish a de- 



CONSTANTS, VARIABLES AND FUNCTIONS. 19 

pendence between the three coordinates of its extremities, 
so that one only is independent. 

The same result will follow if a system of polar coordi- 
nates is used. 

19. Convexity and Concavity. — The side of an arc of 
any curve upon which adjacent tangents, in general, inter- 




sect is called the convex, and the other, or that upon which 
adjacent normals intersect, the concave side. A curve, at 
any point, is said to be convex towards the convex side 
and concave in the opposite direction. 

20. Graphic Representative of a Function of a Single 
Variable. — The relation between any function and its vari- 
able may be expressed by the equation formed by placing 
the function equal to a symbol. Thus, placing /jc equal to 
J, we have J = fx, which expresses the relations between j 
and X, and therefore between the function fx and its 
variable x. y — fx is also the equation of a locus, the 
coordinates of whose points bear the same relations to each 
other as those existing between the corresponding states of 
the function and variable. Therefore, by constructing, as 
in Analytic Geometry, any point of this locus, its oj^dinate 
will represent graphically the state of the function corre- 
sponding to the state of the variable similarly represented 
by its abscissa. The locus thus determined is called the 
graph of the function. It is important to notice that it is 



20 



D IFFEREN TIA L CA LCUL US. 




the ordinate of the graph, not the graph itself, that repre- 
sents the function. 

To illustrate, let the line AB be the 
graph oi fx. Then the ordinate PA is the 
graphic representative oi fx, corresponding 
to a value of x represented by OF. Sim- 
ilarly, /^'^ .represents /x when x — OP' . 
The ordinates PM and P M' of the 
graph MQM' represent two different 
states of the function corresponding to 
the same value of the variable, § 5. 

The ordinates PM, RN, and SO of 
the graph MNO represent equal states 
Y of the function corresponding 

to different values of the vari- 
able, § 5. 

The graph of a function 
which is of the first degree with 
respect to its variable is a 
right line, otherwise not. 

The graph of a continuous function is a continuous line. 

21. Surfaces. — Any portion of any surface may be con- 
sidered as generated by the continuous motion of a line. 
The form of the line and the law of its motion determine 
the nature and class of the surface generated. 

22. Let u represent the area of a varying portion of the 
surface generated by the continuous ^ M 
motion of the ordinate of any given 
line in the plane XY. 

u depends upon the coordinates of 
tlie variable extremities of that portion 
of the mven line which limits it, and varies with each ; but 




CONSTANTS, V^ARIABLES AND FUNCTIONS. 21 




the equation of the given line establishes a dependence be- 
tween these coordinates. Hence, z^ is a function of but one 
independent variable. 

23. Let r=f{v) be the polar equation of any plane 
curve, as DM, referred to the 
pole F, and the right line FS. 
Let u represent the area of a 
varying portion of the surface, 
generated by the radius vector 
revolving about the pole, u will 
change with v and r; but v and r 
are mutually dependent. Hence, 
u is 2i function of but one independent variable. 

24. Let any line in the plane XY, as AM, revolve about 
the axis of X. It will generate a sur- 
face of revolution. 

The same surface may be generated 
by the circumference of a circle, whose 
centre moves along the axis X, with its 
plane perpendicular to it ; and whose radius changes with 
the abscissa of the centre of the circle, so as to always 
equal the corresponding ordinate of the curve AM. The 
radius of the generating circumference is, therefore, a func- 
tion of the abscissa of its centre. Hence, the generating 
circumference, and any varying zone of the surface gener- 
ated as described, is a function of but one independent 
variable. 

25. The area of any surface with two independent vari- 
able dimensions is a function of two independent variables. 
For example, the area of any rectangle with variable sides, 
parallel respectively to the coordinate axes X and Y, is a 
function of the two independent variables x andj>. 




22 DIFFERENTIAL CALCULUS. 

26. Having any surface, as ATL, let A BCD = ?/ be a 
portion included between the coordinate planes XZ, YZy 
and the planes DQR and BPS, parallel to them respec- 
tively. Let OP — X and OQ^^yht independent varia- 




bles, u will depend upon x^y, and z\ but the equation of 
the surface makes z dependent upon x and j. Hence, u is 
a function of but two i7ide pendent variables. Similarly, it 
may be shown that any varying portion of the surface 
included between any four planes, parallel two and two, to 
the coordinate planes XZ and YZ^ is a function of but two 
i7idependcnt variables. 

27. Graphic Representative of a Function of Two 
Variables. — Placing any function of two variables, as/(x jf), 
equal to z, we have z =/(x, j) which expresses the rela- 
tions between the function and its variables. 

The locus whose equation is z — f[x,y), is called the 
grai)hic surface of fi^x^)'), for the reason that the oj^dinaie 



• CONSTANTS, VARIABLES AND FUNCTIONS. 23 

of any of its points will represent graphically the state of 
the function corresponding to the states of the variables 
similarly represented by their respective abscissas. 

It is important to notice that it is the ordinate of the 
graphic surface that represents the function, and not a 
portion of the surface as in the case described in § 26. 

The graphic surface of a function which is of the first 
degree with respect to each of two variables is a plane, 
otherwise not. 

28. Volumes. — Any portion of any volume may be con- 
sidered as generated by the continuous motion of a surface. 
The form of the surface and the law of its motion deter- 
mine the nature and class of the volume. 

29. Let any plane surface included between any line in 
the plane XY^ as AM, and the axis of X be revolved about 

X. It will generate a volume of revo- y M 

lution. The same volume may be gen- 
erated by the circle, whose centre 
moves along the axis X, with its plane 
perpendicular to it ; and whose radius 
changes with the abscissa of the centre of the circle, so as 
to always equal the corresponding ordinate of the curve 
AM. The radius of the generating circle is, therefore, a 
function of the abscissa of its centre. Hence, the generat- 
ing circle, and any varying segment of the volume generated 
as described, is a function of but one indepe7ident variable. 

It is important to notice in this case, that the generating 
surface is limited by the ordinates PA and P' M, corre- 
sponding to the extremities of the limiting curve, which 
ordinates are perpendicular to the axis of revolution. 

30. Having any volume, as ATL, bounded by a surface 
wiiose equation is given, and the coordinate planes, let 




24 



D IFFERENTIA L CAL CUL US. 



ABCD-ON ^=- F be a portion included between the coor- 
dinate planes XZ, YZ, and let the planes DQR and BFS 
be parallel to them respectively. 

Let OP ^ X and OQ=y be independent variables. 
V will depend upon x, y and z ; but the equation of the 
surface makes z dependent upon x andjj;. Hence, F is a 
function of but two independent variables. 




In a similar manner it may be shown that any varying 
portion of the volume included between any four planes, 
parallel two and two, to the coordinate planes XZ and FZ, 
is a function of but two independent variables. 

3I» Any volume with three independent variable dimen- 
sions is a function of three independent variables. For 
example, the volume of any parallelopipedon with variable 
edges parallel, respectively, to the coordinate axes X, Y 
and Z, is a function of x^y and z ; all of which are inde- 
pendent. 



PRINCIPLES OF LIMITS, 2^ 



CHAPTER II. 
PRINCIPLES OF LIMITS. 

32. The Limit of a variable * is a fixed finite quantity or 
expression which the variable, in accordance with a law of 
change, continually approaches, and from which it may be 
made to differ by a quantity less numerically than any 
assumed quantity however small. 

Thus, any constant, as C, is the limit of any variable, as 
/(^), when, under a law, /(;t:) approaches C to within less 
than any assumed value however small it may be. 

Various symbols are used to indicate a limit under a law. 
Thus, assuming that f{x) approaches a limit C as x 
approaches a, we write 

limit /(^) = Lt. f{x) = \imf{x) = limit /(j*:) = C. 

Each form is read, " the limit oi f{pc) as x approaches a."" 

Any variable which under a law approaches zero as a 
limit is called an infinitesimal. Thus, 

1^"^^^ [i - cos ^1 = o. 

Any variable which under a law can exceed all assumed 
values, however great, is called an infinite. It is not a defi- 
nite quantity. 

* In this chapter the term variable is used in its general sense (§ i). 
and includes all functions of variables. 



20 DIFFERENTIAL CALCULUS, 

An infinite cannot be a limit. Thus, 

is a form indicating that as x approaches zero, i/x is un- 
limited. 

A tangent to any curve is a limiting position of a secant 
through the point of tangency, under the law that one or 
more of its points of intersection with the curve approach 
coincidence with the point of tangency. 

In some cases, due to the form of the function or to the 
law of change, the variable can never become equal to its 
limit. Thus, 

limit 



limit ^+i^ii^_r+_i1 

X "4~ I 
But % I, for all values of x.^ 



The circumference of a circle is the limit of the perim- 
eter of an inscribed regular polygon as the number of its 
sides is continually increased. The radius is the limit of 
the apothem, and the circle that of the polygon, under the 
same law. 

An incommensurable number is the limit of its successive 
commensurable approximating values. Thus, the terms of 
the series 1.7, 1.73, 1.732, etc., taken in order, are approach- 
ing 1/3 as a limit. 

In all cases, whether a variable becomes equal or not to 
its limit, the important property is that their difference is 
an infinitesimal. 

An infinitesimal is not necessarily a small quantity in any 
sense. Its essence lies in its power of decreasing numeri- 



PRINCIPLES OF LIMITS. 27 

cally ; in other words, in having zero as a limit, and not in 
any small value that it may have. It is frequently defined 
as " ail infiiiitely small quantity "y that is not, however, its 
significance as here used. 

In representing infinitesimals by geometric figures they 
should be drawn of convenient size ; and it is useless to 
strain the imagination in vain efforts to conceive of the 
appearance of the figure when the infinitesimals decrease 
beyond our perceptive faculties. Usually one or two auxil- 
iary figures representing the magnitudes at one or two of 
their states under the law give all the assistance that can 
be derived from figures. 

In all cases, when referring to the limit of a variable, it is 
necessary to give the law ; for the limit depends not only 
upon the variable, but also upon the law by which it 
changes. Under a law, a determinate variable has but one 
limit ; but it may have different limits under different laws. 

An important consequence of the definition of a limit is 
that if two variables, in approaching limits under a law, 
have their corresponding values always equal, their limits 
will be equal. Thus, for all values of x^ we have 

(c^ — x^^/(a — x) =^ a -{- Xf 
hence 

J™'i («'- ^')/('^ - *) = lim [a + x]= za. 

33 • A variable which^ in approaching a limit ^ ultimately has 
and retains a constant sign cannot have a limit with a contrary 
sign. 

For suppose /(j^) becomes and remains positive, and that 
limit f{x) — — C. From the definition of a limit, f(pc) 
may be made to differ from — C by a value numerically 



28 DIFFERENTIAL CALCULUS. 

less than C. It would therefore become negative, which is 
contrary to the hypothesis. In a similar manner, it may be 
shown that a variable always negative cannot have a posi- 
tive limit. 

34* If the difference between the cori^esponding values of 
any two variables, approaching limits^ is an infinitesimal^ the 
va: iables have the same limit.^ 

Let U and V represent any two variables giving 

U- V= S, or U= V+d, 

in which S is an infinitesimal. 

Let Cbe the limit of C/^, then U =■ C — e, in which e is 
an infinitesimal. 

Substituting we have 

C:-e=F+d, or C - V = d ^ e, 

the second member of which is an infinitesimal. Hence, C 
is the limit of V. 

35. The limit of the sum of any finite number of variables 
is the sum of their limits. 

Let U^ — V, JV, etc., represent any variables, and A, 
— B, C, etc., their respective limits ; then 

l7=A-€, - F= -B+S, IV=C- Go.'etc, 

'n which e, d, go, etc., are infinitesimals. 

Adding the corresponding members we have 

C/~ F+lV-{- etc.= A- B -rC-^etc- e + (^- &? + etc. 

* In order to avoid the frequent repetition of the expression '" under 
the law," it will be assumed, unless otherwise stated, that the chancres 
in all the variables considered together, or in the same discussion, 
are due to one and the same law; that all variables and their functions 
are continuous between all states considered, and that they have 
limits under the law. 



FEIiVCIFLES OF LIMITS. 2g 

Hence, 

limit [l/-V+JV-^etc.] ^ A -^ + C+etc. 

= \\mU — lim F+ lim W-\- etc. 

36. In general, the limit of the product of any two variables 
is the product of their limits. 

Let U and V represent any two variables having A and 
B^ respectively, as limits. 

Then U — A — e and V= B — S, in which e and d 
are infinitesimals. Multiplying member by member, we 
have 

C/V=AB-Be-AS-^ed, 
and 

limit [CrV] = AB = limit Cr limit F, 

It follows that, in general, the limit of any power or root 
of any variable is the corresponding power or root of its limit. 

Thus, limit U" = (limit UY, and limit U~^ — (limit t^)«, 

Having a"" = N, x and N approach corresponding limits 

together; hence ^""^-^ = lim JV = lim a"", and lim x = 

log lim JV. Also, since x = log JV, we have lim x = lim 

log JV. Therefore log lim JV = lim log JV. 

37. In general, the limit of the quotient of any variables is 
the quotient of their limits. 

With the same notation as in § 36, we have 

limit ^ = lim lUV-'\ = lim ^7[lim F]- = '^^^ = |. 

When B = o, and A ^ o, U/V'is unlimited. 
When ^ = o = ^, the principle fails to determine the 
limit which by definition is determinate, 

38. It follows from §§ 35, Z^, 37, that, in general, the 



30 



D IFFEREN TIA L CALC UL US. 



limit of any function of any variables is the same functio7t of 
their respective limits. 
Thus, in general, 

limit /(^ V, ) =/(Iim U, lim F, ), 

and to obtain the limit of any function of variables we, in 
general, substitute for each variable its limit. ' 

39. Exceptions to the above general rule arise, and are 
indicated by the occurrence of some indeterminate form, as 

0/0, CO /oO , OCO , 00 — 00 , 0°, 00 °, I*. 

To illustrate, having /(:<;) = {pc"— i)/{x — i), the general 
rule gives limit /(^) = 0/0, whereas we find 

/(2) = 3> /(i-5) = 2.5, /(i.i) = 2.i, /(i.oi) = 2.01, 
/(i.ooi) = 2.001, etc., 

and the nearer we take x to i, the nearer will /(:^) approach 
to 2. By taking jc sufficiently near to i,/(^) may be made 
to differ from 2 by a number less numerically than any as- 
sumed number however small. Hence, 2 is (§ 2^2) the limit 
oi fi^x) as x-Wf-^Y. It should also be ^^bserved that 2, con- 
sidered with the states of /(^) which immediately precede 
and follow it, conforms to the law of continuity. 



Y 




^ 


M' 






w 


^ 


^;=^=^ 


A?/ 






,y 








/b/ 


/ 


y 




y 




^^ 




p 


AX 


p' 


X 



H 

To illustrate a failing case geometrically, let the curve 
BMM' be the graph of a function. Take any state, as 



PRINCIPLES OF LIMITS, 3 1 

PJ^ corresponding to ^ = OP, and increase x by PP' rep- 
resented by L.X. Draw the ordinate P' M' and the secant 
MM'. Through M draw M Q' parallel to X. Q'M', de- 
noted by Ay, will represent the increment of the function 
corresponding to Ajt:. 

Q'M'/PP' = Ay/ Ax = tan Q'MM' will be the ratio 
of the increment of the function to the corresponding in- 
crement of the variable. 

At J/ draw MX tangent to the curve. Then, under the 
law that Ax approaches zero, the secant MM' will ap- 
proach coincidence with the tangent MT, and the angle 
Q'MM' will approach the angle Q'MT, or its equal XHT, 
as a limit. 

Hence 

limit {Ay/ Ax) — lim. tan Q'MM' = tan XHT, 

whereas the general rule gives o/o as a result. 

We observe from the above illustration that tAe limit of 
the ratio of any increment of any function of a single variable 
to the corresponding increment of the variable, wider the law 
that the increinent of the variable approaches zero, is equal to 
the tafigent of the angle made with the axis of abscissas by a 
tangenty to the graph of the function, at the point correspond- 
ing to the state considered. 

When J/' coincides with J/ the secant may have any one 
of an infinite number of positions other than that of the 
tangent line MT, for the only condition then imposed is 
that it shall pass through M. 

Therefore, while limit (Aj/Ajc) is definite, and equal to 

the tangent of the angle that the tangent line at J/ makes 



32 DIFFERENTIAL CALCULUS, 

with X, limit Aj/limit A^ = o/o indicates that the tangent 
of the angle which the secant makes with X becomes inde- 
terminate when M' coincides with M. 

Limit (Aj/Aj^) is, therefore, one of the many values that 
limit AJ^^/limit l\x may have under the law. 

The exceptional cases, in general, require transformation 
in order that factors common to the numerator and de- 
nominator may be cancelled, or from which the limit may 
otherwise be determined. They are of the highest impor- 
tance, for the Differential Calculus, as it will be seen, is 
based upon the limit of the ratio of the increment of the 
function to the corresponding increment of the variable 
under the law that the increment of the variable vanishes. 
The remainder of this chapter will, therefore, be devoted to 
certain important exceptional cases and methods. 

40. ^^^^^ :sr--j^ ^ my— ^ 

x'm-^y X — y 

This formula is deduced in Algebra for all commensu- 
rable values of m. Since (§ 32) any incommensurable 
number is the limit of its successive commensurable ap- 
proximating values, the formula holds true when m is 
incommensurable. 

Limit ^_Q^ a^ j aP^_a_ 

As X »^^ CO , it reaches a value k > a, thereafter 



< 1; also 1 — 

i \x 



< 



«" la\ 

kW 



but 



limit ^ll±Y~"^Q . limit ^ ^ q. 



PRINCIPLES OF LIMITS. 



33 



42. Limit(i + yy'y — e. 



2 

(i-;;)(i-2j;). . . [i-(^-iM ^^^ 

As j^' B-^ o each term approaches, as a limit, the corre- 
sponding term of the series 

i + i + ^4-,-^ + ...+ l^ + etc., 

the sum of which is shown in Algebra to be ^ = 2.71828 . . . 
43. Limit a^ — I 



h-^-^o 



log a. 



Place a!" = 1 + J, whence h = loga(i + ^), and j/b-^/^m^o ; 
giving 



limit 



limit 



y 



h^o h ~-^^°log.(i+j) 

T 

= lim 



log^. 



log.(i + yY'y \ogae 
44. If unity is the limit of the ratio of any two -variables, 

the limit of any function of one will be equal to the limit of 

the same function of the other. 

Let C/" and ^represent any two variables, giving X\v^\iU/V 

= I. Then 



limit/(C^) = lim/[^]=/( 



lim 
= /(lim F) = lim/(r). 



V 



lim V\ 



34 DIFFERENTIAL CALCULUS, 

Thus, 

lim[C:+ VX = lim [C -f ^] ; lim {CU^, = lim [CV} 

lim C ^ = lim C^ ; lim [ U/ W] = lim [ V/ IV]. 

Therefore, in searchi?ig for the limit of any function under 
a law, we may replace any variable entering it by another vari- 
able, provided that, under the satJie law, unity is the limit of 
the ratio of the two variables interchanged. The great ad- 
vantage in so doing arises when it enables us to determine 
the required limit more readily. Thus, in the last example 
above we may be able to determine the limit of F/^more 
readily than that of U/W. 

In making the above substitution it is important to notice 
that the limits only are equal, and that corresponding values 
of the quantities interchanged, in general, are not equal to 
each other. 

The privilege of replacing one variable by another under 
the conditions described, so facilitates the determination 
of limits in certain exceptional cases, that it is important to 
determine under what circumstances the limit of the ratio 
of two variables is equal to unity. 

45* In general when lim £/"— lim F, 

lim lU/VX^Xim U/Xim V= i. §37. 

That is, in general, unity is the limit of the ratio of any 
two variables when, under the same law, they have the same 
limit, or, what is equivalent, when the difference between 
their corresponding values is an infinitesimal. 

If, however, lim 17=^ lim F = o, it does not follow that 
lim [CI/ V] = I {§ 39). Such cases require special investi- 
gation, and the following are selected on account of their 
subsequent impoitance. 



PRINCIPLES OF LIMITS. 



35 



\ 


M 


Q' 


Q 


V 


m' 


P 


AX 


P' ^ 



Q ^ 


' 


; 


L.^^^"^ 


P' X 


P Aa^ 



46. Take any plane surface, as PMM'P\ included be- 
tween any arc, as MM', the ordinates of its extremities, 
and the axis of X, 

Through J/ and J/', Y 
respectively, draw MQ' 
and M'Q parallel to 
X, and complete the 
rectangle MQM'Q'. 
Let y = PM, and y' = P' M' . 

Then as PP'= Ax, b-> o, we ultimately have 

PQM'P''%PMM'P%PMQ'P\ 

and y'M-^y. 

Therefore ^^ IML^l = n^ IA^ = , 
and limit \PMM' P' / PMQ' P'\= i. 

Hence (§ 44) 

limit PMM'P ' ^ j.^ PMQ^P ' ^ ^^^y_Ax _ 
Ax:^->o A^ Ax Ax 

If the coordinate axes make an angle with each other, 
then 



=y- 



limit 



PMM'P' ,. ysin Bax . ,. 

= lim =y sm c/. 



Ax 



Ax 



47« Let MPM' be the surface generated by the radius 
vector PM = r, revolving about 
P, as a pole, from any assumed 
position, as PM, to any other, as 
PM'. Let Av represent the 
corresponding angle MPM' . 
With /^ as a centre, and the radii 
PM and P M' , describe the arcs 
MQ' and M' R respectively. 




36 DIFFERENTIAL CALCULUS. 

•Then, as Az^ ^W)-^ o, we ultimately have, in any case, 
area RPM' ^ area MPM' % area MPQ% 
and limit [area RPM'Jzx^d. MPQ'] = i. 

Hence, I'^^o [area MPM'/sirGSi MPQ'] = i. 
Therefore (§ 44) 



limit z= lim ^^ = hm — 



A57B^-^0 



l\V 



/\V 



Av/2 __ r^ 
/\v 2 ' 



48. Let FMM' P' be any plane figure as described in §46, 

and MQM'Q' the cor- 
B responding rectangle. 

Q' Revolve the entire 

M^ figure about X. 

„/ Then as Aj\;b->o, we 

r — ^x 



M 



P Ace 



.M 



Pax 



ultimately have 

Vol. gen. by > Vol oren. by > Vol. gen. by 
PQM'P' < PMM'P' < PMQ'P\ 



But 



limit [ Vol. gen by /VoL gen. by~| 
Ax:^o L PQM'P' I PMQ'P' J 



lim 



ny^ Ax __ 
TZy^ Ax 



Hence, 



limit FVol. gen. by /Vol. gen. by"1_ 
\x^^o L PMM'P' I PMQ'P' _\- ^' 



Therefore (§ 44) 



.. . /Vol. gen. by\ /Vol. gen \ 

limit / pMM'P' . I PMQ'P' ) 

\ AX / ^ Ax ^ 



lim \ny^ Ax/ Ax\ = Tty^, 



PRINCIPLES OF LIMITS. 



37 



49. Let MJVM'N' be a portion of any surface included 
between the coordinate planes ZX\ ZY and the two planes 
N'SE and NP D parallel to them respectively. 

Denote the corresponding volume MNM'N\ OFFS 
by V. Construct the parallelopipedons OFFS-OM and 




OFFS-FM', and represent their volumes by F and F' 
respectively. Let OF = h, OS = k, OF :=^ /, OM = z, 
and FM' = z' . 

Then as h^m^k-w^ o, or what is equivalent, as /»^-» o, 
whence 2' ^^2r, V will, in any case, ultimately be, and 
remain, between F and F' . 



But 



Hence, 



limit .p/pr 
limit 



limit 



\.VlF-\ 



Izhk/z'hk] = I. 
and (§ 44) 



J^o [^Aect. (9/^7^^] = lim [/^/M] = liiv 'zhk/hk\ = z. 



^8 



DIFFERENTIAL CALCULUS. 



50. Let MNM'N', denoted by S, be a portion of any 
surface included between the coordinate planes ZX, ZY 
and the two planes N' SE and NPD parallel to them 
respectively. 

Let OP = h, and OS = k. 

At M draw the tangents MB and MB' to the curves 
MN and MN' respectively, complete the parallelogram 
MBQB', and denote it by T. T is the portion of the 
tangent plane to the surface at M included between the 




planes which limit S. Let /5 equal the angle which T 
makes with XY, giving T cos (3 = OPFS. Inscribed 
in S^ conceive an auxiliary surface, composed of 11 plane 
triangles the sum of which, as ;/ ^-^ co , will have 6" as a 
limit and such that the sum of their projections upon XY 
will equal OPFS, The two triangles MN' M' and MNM' 
in the figure illustrate a set fulfilling the conditions. Let /, 
/', etc., represent the areas of the triangles and ^, ^', etc., 
the angles which their planes respectively make with XY. 



PRINCIPLES OF LIMITS, 39 

Then OPFS = ^/ cos ^ = r cos /?, 

and :Et cos 0/T cos P = 1 (i) 

As ;z ^-> 00 , S remaining constant, each triangle is an 
infinitesimal, and we have 

The same effect and result follows if -5" is made infini- 
tesimal and n remains constant. Hence, under the law 
that A and k vanish, or, what is equivalent, that OJ^y repre- 
sented by /, is infinitesimal, we have 

^[s/2^] = ^ W 

Under the same law, /3 is the common limit of 6, 6\ etc. 
Hence (i) 

jl^'^ l^t cos e/T COS /?] = lim [^f/T] = i. . (3) 

Therefore (2), (3), ^^\[S/ T] = i, and (§ 44) 

^ Ji'^jLo [^M^]= li«^ i^M^I = lim [{M/cos (5)/hk\ 

— I /cos ft. 

51 • Unity is the limit of the ratio of an angle to its si?ie, of 
an angle to its tangent^ and of the tangent of an angle to its 
sine, as the angle approaches zero. 

Let OCM = given in radians 
be any angle less than 71/2 ; then 
tan > > sin 0, and as ;^-> o, 
we have always C 

tan ^ 

sm sin 

B„j limit ta,L0 ^ ,i„^ -J- = I. 

<*s^o sm <?> cos <p 




40 DIFFERENTIAL CALCULUS 

Hence, since 0/sin <p is always between tan 0/sin and 
unity, we have 

!^"^^^ [0/sin ] = I. 

Similarly, since i > 0/tan > sin 0/tan 0, we have 

52. U'm'fy IS the limit of the ratio of any arc of a7iy curve 
to its chords as the arc approaches zero. 

Let s denote the length of any arc of any curve, and 
conceive it to be divided into n equal parts, the consecutive 
points of division being joined by chords forming an in- 
scribed broken line whose length is designated by/. 

Then }:t\^IP\ = ^. 

Under the above law the equal arcs of s vary inversely 
with n\ hence the same effect and result may be caused by 
retaining any fixed value for «, and making s approach 
zero. Hence, 

and if «= I, J^^^Jarc/chord] = i. 

53. Unity is the limit of the ratio of the surface generated 
by any arc of any curve under a law to that generated by its 
chord as the arc approaches zero. 

Let s^p and n denote, respectively, the same quantities 
as in the last article, and conceive s and;> to move together, 
under a law, so as to generate two surfaces represented by 
S and P respectively. 

As ^/«^^oo, any state of s without regard to form is the 
limit of the corresponding state of/, and 6" is the limit oi P. 



PRINCIPLES OF LIMITS. 



41 



That is, 



limit 



ls/p-\ = I. 



As ;2B-^oo, the equal arcs of s approach zero; hence the 
same effect and result may be caused by retaining any fixed 
value for 7i and making ^:^^o. Therefore 



limit 



Vs/P^ 



\in 



limit 



Sur. gen by an arc 



'^^^ LSur. gen. by its ch 



re ~j 



The results determined in §§ 51, 52 and 53, with the 
principle § 44, are of great value in finding the limits in the 
following exceptional cases. 

54. \.^\MM'~s be any arc of any plane curve, T^il/ 
and F'M' the ordinates of its extremities. 









1 


r 




Y 


,>C 


L' 






M 


^ 


^^=^^ 


., 






Y 








^^/ 


V 


V 




'n' 




y^ 




p 


Ax 


p' 


X 



H 

Through J/" draw the chord MM' = c, the tangent MT 
= 3, and MQ'= FP'— Ax, parallel to X. 

1-1 .^.^/^ 1 ^ s'mMM'T 

From the triangle MM' T, we have — = -: — TT-r^TTi' 

^ ' ^ sm M7M' 

As Ajic approaches zero, the arc s and the angle M'MT 

vanish, but the angle T remains constant. Hence, the 

angle J/J/T approaches [180"— T], and 

'b 



limit 



^ limit fsin^M^I ^ sin_( 
j«^o |_ sin T J ! 



80°- r) 
-■ — ^= — ^ — I. 

sm 1 



42 



D IFFEREN 1 'I A L CALC UL US. 



Hence, since (§ 52) 

lr.':^A] = i, we have (§44) iH [V^] - l 
Therefore (§ 44) 

b ,. FaVccs Q'MT'^ I 



iri-ii=«-^=i.».L- 



] 



t^x J COS Q'MT 

55* Let r ^ f{v) be the polar equation of any plane 
curve, as AMM\ referred to the ri^ht line PD, and 
pole P. 




Let AM = ^ be any portion of the curve, and PM = r 
the radius vector corresponding to M. 

Regarding ^ as a function of v (§ 18) let v be increased 
by MPM'—ixv. The arc MM' will be the correspond- 
ing increment of s. Draw MQ' perpendicular to PM\ 
and denote PM' by r' . Then (§§ 44, 52) we have 



limit arc MM ' _ ch. M M' 



Av 



Av 



lim 



/mQ'" 4- Q'jW 



_ . i /(^ sin A e')"^ +(/—;- cos AvY 



PRINCIPLES OF LIMITS. 



43 



Hence (§ 51) 






We also have 



tan FMD = }''^'^ tan Q'M'M=\im ^-^, 



t\Vl 



= lim -7 = lim —. 

r — r cos Av r — t 



Hence, 



limit r - r 



which, substituted in (i), gives 



limit arc M M' ^ ^ ^, _^ 



Az'B^O AZ^ 



tan' FMB 



If the radius vector PM coincides with the normal to 
the curve at M, we have 

tan FMI? = 00 , and }^'^^ [arc MM'/ Av] = r. 

56. Let any plane figure, as j 

PMM'P\ included between any 
plane arc, as MM'\ the ordi- ^ 
nates of its extremities and the 
axis of X, be revolved about X. 
Then (§§ 44, 53) 
limit Sur. gen. by arc MM' / 

= lim ^^""^ ^^'^' ^^ ''^' ^^' ^ lim "^^-^ ^^'^'^ 
Ax Ax 

__ • '^{y + y ) ^ Vcos QMM* _ 2 ny 

"" A^^ cos (2'JO^* 




44 DIFFERENTIAL CALCULUS. 



CHAPTER III. 

RATE OF CHANGE OF A FUNCTION. 

57* A function changes uniformly with respect to a vari- 
able when from each state all increments of the variable 
are directly proportional to the corresponding increments 
of the function. 

It follows that from all states equal increments of the 
function correspond to equal increments of the variable; 
also that the ratio of any increment of the function to the 
corresponding increment of the variable is constant. 

Thus, having 2ax^ increase x by any amount denoted by 
h, then 2a{x -\- K) — 2ax — 2ah will be the corresponding 
increment of 2ax, It varies directly with h, it is the same 
for all states of the function, and 2ah/h = 2^ is a constant. 
Hence, 2ax changes uniformly with respect to x. 

'Let fx be any uniformly varying function, and /i any in- 
crement of X. /{x -\- /i) — fx will be the corresponding 
increment of the function, and 

[/(^ + ^) — fx\/h = constant = A. 

Hence /(x -\-A)=A/i-{- f{x), 

in which .r = o, gives f{h) = Ah -\- /(o). 

Therefore, fx = Ax +/(o). 



RATE OF CHANGE OF A FUNCTION. 



45 



Hence, all functions which change uniformly with respect 
to a variable are algebraic, and of the first degree with 
respect to that variable, and all algebraic functions of the 
first degree with respect to a variable change uniformly 
with that variable. 

The graphs of such functions are right lines, and any 
function whose graph is a right line changes uniformly 
with respect to the variable. 

To illustrate, the right line AD is the graph of a function. 
Consider any state, as that represented by the ordinate PA, 







c 


c 




B 


/ 


Q" 
s" 

P" 


A 


y^ 


Q' 




^ 


P 


p' 


?'" X 



Increase the corresponding value of the variable, repre- 
sented by OP, by any increments, as PP' and PP" . Q' B 
and S"C will represent the corresponding increments of 
the function, and the similar triangles AQ' B and AS"C 
give 

AQ; : AS'' :: Q'B : S"C. 



That is, the corresponding increments of the variable and 
function are proportional. 

By giving to jc = OP any equal increments, as PP' , 
P' P" ,P" P"\ in succession, the corresponding increments 



4^ DIFFERENTIAL CALCULUS. 

of the function, Q' B, Q" C, and Q''D, are equal to each 
other. 

It is also evident that the ratio of any increment of the 
function to the corresponding increment of the variable, as 
Q'BjPP', or S"C/FF'\ or 'q"'DIP"F''\ is constant. 

58. Having the function 2.v, — ■ 



"='<x 


gives 


2X =^ 2. , 
<2 


X^ 2 


" 


2X — 4. 


<I 




2X = t.<^ 


^==3 





etc. etc. 

Hence, the function 2x increases two units v/hile the 
variable increases one ; in other words, twice as fast. 
Having the function 5^, — 



x 


= '<! 


gives 


5^: 


=-- 5- 


<5 


X 


= 2 




5^= 


= 10. 






<I 


<< 






<5 


X 


= 3 
etc. 




5^= 


= 15- 
etc. 





Therefore, the function ^x changes fives times as fast as 
the variable. 

In general, different functions change, with respect to 
their variables, with different degrees of rapidity. 

The measure of the relative degree of rapidity of change of 
a function and its variable at any state is called the rate of 
change of the function^ with respect to the variable^ correspond- 
ing to the state. 

A rate of change of a function with respect to a variable, 
corresponding to a state, is an answer to the question: At 
the state considered, how many times as fast as the variable 
is the function changing ? 



RATE OF CHANGE OF A FUNCTION. 47 

59. Since, from all states, any uniformly varying function 
receives equal increments for equal increments of the 
variable, its rate, from state to state, is constant, and equal to 
the ratio of any incre77ient of the function to the correspondiiig 
increment of the variable. 

2(X -\- H) — 2X 

Thus, rate of 2x — = 2. 

K,,e of 3. + 3 --= M^±^^±|l^Il^±^ = 3. 
Rate of 5^ - 3 = ^^^" + ^^ ~ ""} ~ ^^" " '^ - 5- 



It follows that the rate of any uniformly varying func- 
tion is equal to the tangent of the angle which its graph 
makes with the axis of the variable, also that the product 
of the rate by any increment of the variable is equal to the 
corresponding increment of the function. 

60. It follows from § 57 that algebraic functions not of 
the first degree with respect to the variable, and all 
transcendental functions, are, with respect to the variable, 
non-uniformly varying functions. Thus, having ay^ , in- 
crease X by any amount as //. 2axh -j- ah^ , which varies 
with X, is the corresponding increment of ax^. Also the 
ratio {2axh -{- ah'^)/h = 2ax -\- ah =^ (p{x, h). Hence, ax^ 
does not change uniformly with respect to x. 

Having /(jv"), in which n is not equal to i, then [f{x -f- /?)" 
— f{x^)yh = (p{x, h), and/(:v:") is therefore a non-uni- 
formly varying function with respect to x. 

The graphs of such functions are curves, and any func- 
tion whose graph is a curve does not change uniformly 
with respect to the variable. 



48 



DIFFERENTIAL CALCULUS, 



To illustrate, the curve MNT is the graph of a func- 
tion. Increase, in succession, 
any value of x, as 0P\ by any 
amounts, as F' P" and F' R. 
F'F"IF'R = QB/ST is the 
ratio of the increments of the 
variable, and it differs from 
QN/ST, which is the ratio of 
the corresponding increments of 
the function. Hence (§ 57), the ordinate of MNT, and 
the function represented by it, do not change uniformly 











\ 


/ 


Y 




/^ 






/ 


/y 


i 







U 


^ 


^ 


Q 


s 


'^ 


?' 


p" 


R X 



with X. 






61. In the function 


:x'- 




X = "i 
■y : — A 


gives 

it 

a 
li 


2X''= 2. 
2 <I4 



2X =32. 



Which shows that at different states the function 2^* has 
different rates wHh respect to x. 

A non-uniformly varying function, in general, receives 
Unequal increments for equal increments of the variable. 
It follows that its rate varies from state to state. 

Any particular rate is, therefore, designated as the rate 
corresponding to a particular state. 

If a function has two or more states corresponding to any 
value of the variable, each state will have a rate. 

If a function has equal states for different values of the 
variable, it may have a different rate at each ; in which 
case it is necessary to indicate the value of the variable 
corresponding to the state considered. 



RATE OF CHANGE OF A FUNCTION. 



49 



62. Rate of Change of any Function. Let fx be any 
function. Denote by R its rate, 
with respect to x, corresponding 
to any state, as PA. Increase 
X = OP \yy h = PP', and let 
P' represent the rate of the func- 
tion at the new state f{x -f- ^) = 
P'B. QB = f{x ^K)-fx will 
represent the corresponding increment of the function. 

\i fx changes uniformly, 




Ax^h)-fx 



^, = tan QAP 



will (§ 59) be its constant rate, P = P', and its graph will 
be the right line AP (§ 57). 

If fx is a non-uniformly varying function, its graph is a 
curve, as AEB^ and in general P 4^ P' . 

In this case let PP' = /^ be taken so small that while x 
varies from OP to OP' ^ the rate oi fx will either decrease 
or increase in order from P to P\ 

In either case, [/(-^ + h) — foc\/h, which is the con- 
stant rate of the uniformly varying function whose graph 
is the right line AB^ will be between P and P' \ for other- 
wise the uniformly varying function would change between 
the states considered by an amount greater or less than 
QB. 

In other words, \_f{x -^ K) — foc\/h is the rate oi fx at 
some state, as P" E, between PA and P'B. Let PP"= Oh, 
in which B is the proper fraction PP" /PP', 



Then 



P"E = fix + Oh), 



50 DIFFERENTIAL CALCULUS. 

and y(x + Ji) - fx\/h = rate oi /{x + dA). . (i) 
This relation will always exist as h'm-^o. 



Hence, "^"^'^ 



- f{x + ^) - /r l 
7 = rate oi fx. (2) 



That is, //^^ rate of change of any ftmctioji with respect to a 
variable^ corresponding to any state ^ is equal to the limit of the 
ratio of any increment of the function^ from the state consid- 
ered^ to the- corresponding increment of the variable^ under the 
law that the increment of the variable approaches zero. 

The above principle enables us to find the rate of any 
function with respect to a variable, corresponding to any 
state, by the following general rule : 

Give to the variable any variable increment^ and from the 
corresponding state of the functio7i subtract the primitive. 
Divide the retnainder by the increment of the variable, and 
determi?ie the limit of this ratio, under the law that the incre- 
ment of the variable approaches zero. In the result substitute 
the value of the variable corresponding to the state. 

It should be observed that a rate, determined by the 
above method, is equal to a limit of a ratio of two infinitesi- 
mals, which limit is determinate ; and that it is not equal to 
the ratio of their limits. § 39. 

To illustrate, let the curve AB" B' in the figure on page 
51 be the graph oi fx,d,Xi^ let PA be the state at which the 
rate is required. 

When h = FP\ 

f(x-^rh)-fx _ Q'B' _ 

^ — -ppT — tan Q AB , 

which is the rate of the function whose graph is the secant 
AB\ 



RATE OF CHANGE OF A FUNCTION, 
When h = FP", 



51 



pp, 



tan Q"AB", 



which is the rate of the function whose graph is the secant 
AB'\ 

As /^«»-»o, the above ratio is always the rate of a function 
whose graph is a secant approaching the tangent ^T as a 




limit. Hence, the limit of the ratio is the rate of the func- 
tion whose graph is the tangent ^7", and is equal to the 
tangent of the angle Q'AT. 

That is, the limit of the ratio of any increment of any 
functio7i of a single variable to the corresponding increment 
of the variable^ tinder the law that the increment of the vari- 
able approaches zero, is equal to the tangent of the angle made 
with the axis of abscissas by a tangeiit at the corresponding 
point of the graph of the function. 

The same result was obtained in § 39, The angle which 
the tangent AT makes with X is called the inclination, and 
the numerical value of its tangent is called the slope of the 
graph at M. 

The rate of any function at any state is therefore the 
same as that of a uniformly varying function whose graph 



52 



DIFFERENTIAL CALCULUS. 



is the tangent to tjie graph of the given function at the 
point corresponding to the state considered. 

This agrees with previous conceptions and definitions, 
for the direction of the motion of the point generating the 
curve at any position is along the tangent at the point, and 
the ordinate of the curve representing the state considered 
is changing at the same rate as the ordinate of the corre- 
sponding tangent. 

It follows that the product of the rate of a non-uniformly 
varying function by an increment of the variable is not, in 
general, equal to the corresponding increment of the func- 
tion. 

63. Let jj/ — PA represent any state of any increasing func- 
tion of x\ andy the new state corresponding to an incre- 




ment PP' = h, of the variable, {y* — y)/h will be the ratio 
of the increment of the function to that of the variable. If 
h is assumed sufficiently small, this ratio will be positive and 
remain so as /^^-»o. § 15. 



Hence (§33), 



limit 



[(y ~ y)l^ — tan XEA is positive. 



Let y = Pi^i represent any state of a decreasing function ; 
and y' its new state due to an increment of the variable 
equal to P,P/ = h. Then (/ — y)lh will be negative, if h 
is small enough, and will remain so as /^ ^^-» o. 



Hence, 



limit 

hWHfO 



\ky' ~ y)/^^] — ta^ XE^A is negative. 



J^A TE OF CHANGE OF A FUNCTION. 



53 



Therefore, the rate corresponding to any state of aji mcr eas- 
ing fu nctio7t is positive, and of a decreasing function is 7iega- 
tive. 

It follows that a function is an increasing one when its 
rate is positive, a?id a decreasing one when its rate is negative. 

When, for any state of a function, as the one represented 
by PA or P' A\ the function is neither increasing nor de- 



creasing, its rate is zero, and the tangent at the correspond- 
ing point of its graph is parallel to the axis of x. 

If for any state of a function, as the one represented by 
P"A", the rate is unlimited, the corresponding tangent to 
its graph is perpendicular to the axis of x. 

EXERCISES. 

Find the rate of change of each of the following functions: 



I. 2ax. 



2. X\ 



3. ax^ -f- bx. 



limit Y 2a{x - ^ h) — 2ax 
h 



A°^-i^;L H ""' Y"'- 



Ans limit r(^^_)lz-_f.n=^, 
h-M-^o\ h J 

(a a\ 

— TT ~ \ ^ 



f5. lax\ 



Ans 



Ans i\ax. 



6. x^. 



Ans. "ix"^. 



54 DIFFERENTIAL CALCULUS. 



7. ^x^. Ans. \t>o<;'^. 8. — ■ — Ans. — 



i-\-x ■ {i-\-xf 



Ans. 



3+^ (3 -f-^y 



10. How is the ordinate of a parabola, corresponding to 

^ = 3, changing with respect to the abscissa ? 



Rate of 



_ . limit yV^Pi^-^ ^) - V^px~^ , ,— [~(x-^/i)^ - x^~] 

y-^ /,^o [_ \ J -±|/2/z[_ ^ J 

11. Same corresponding to focus? Ans. i. 

12. Find the abscissa of the point, of the parabola y = 
4x, where the ordinate is changing twice as fast as the 
abscissa. Ans. x = 1/4. 

13. At the vertex of a parabola, how is the ordinate 
changing as compared with the abscissa ? 

14. Find the rate of change of the abscissa of a parabola 



with respect to the ordinate ? j\ns. y/p = ± Vix/p. 

15. Find the coordinates of the point of the parabola 
y = 8x, where the abscissa is changing twice as fast as the 
o'-dinate. Ans. (8, 8). 

16. Find the rate of change of the ordinate of the right 
;ine 2y — -^x ^= 12, with respect to the abscissa. Ans» 3/2 

17. A point moves from the origin so that y always in 
creases 5/4 times as fast as x\ find the equation of the line 
generated. 

5/4 = tan of angle line makes with X. .' . Ans. ^,y — 5x. 



RATE OF CHANGE OF A FUNCTION. S5 



i8. Find the slope of the graph of ± r i2jc when x = 
1/2. Ans. ±2.4495. 

19. Find the abscissa of the point of the graph of V 2px 
when the slope is i. Ans. x = p/2. 

20. Find the angles which the lines y^ =■ 2>x, and 3jj^ — 
2X = 8, make with each other at their intersections. 

Ans. 11° 18' 35", and 7° 7' 30". 

21. Find the angles which the lines y^ = /^x, and 2y = 
X -]- 2, make with each other at their intersections. 

Ans. 10" 14', and t,7,° 4'. 

64. A function of two or more variables is a uniformly 
.varying function with respect to all of its variables when 
it changes uniformly with respect to each. It follows 
(§ 57) t^"^^t ^11 uniformly varying functions are algebraic, 
and of the first degree with respect to each variable, and all 
algebraic functions of the first degree with respect to each 
variable are uniformly varying functions. 

Let u = Ax -\- By -\- Cz -{- etc., in which A, B, C, etc., 
are constants, be any uniformly varying function. Increase 
the variables x, y, z, etc., respectively, by any increments, as 
Af k, /, etc., giving a new state, 

u' = A{x + /^) + B{y-{-k) + C(z-i-/) + etc. 

u^ — u = Ah + Bk -\- CI -\r etc., is the corresponding in- 
crement of the function. It is independent of the state of 
the function, dependent upon the increments of the variables, 
and is equal to the sum of the increments due to the 
increase of each variable separately. 

65. A uniformly varying function of two variables is 
some particular case of the general expression Ax + By + 
C^ in which A^ B and C are constants. Its graphic surface 



56 



DIFFERENTIAL CALCULUS. 



is a plane (§ 27), and any function of two variables whose 
graphic surface is a plane is a uniformly varying function. 
To illustrate, take any ordinate, as NM^ of any plane, as 
MLH. Through NM pass the planes MNR and MNP 
parallel, respectively, to ZX and ZF, intersecting the given 
plane in the lines J/ZTand ML. Assume iVi? as the incre- 
ment of Xy and NF as the increment of y. Complete the 




parallelogram NS, the parallelopipedon NQ, and in the 
given plane the parallelogram MLTH. Produce RK to ZT, 
SQ to T, and PA to L. KH is the increment of MN due 
to the increment, NR, of x alone, and AL is its increment 
due solely to NP^ the increment of ji^. ^T^is the entire 
increment of MN due to the increase of both variables 
together. 

Draw HD parallel to KQ, and draw AD\ it will be par- 
allel to LT, because it is parallel to J/ iT, which is parallel 
toZr. Hence, i?r= ^Z. 



FA TE OF CHANGE OF A FUNCTION. 57 

Therefore, QT=QD-\-nT= KH ^ AL. 

That is, the total increment of any uniformly varying 
function of two variables from any state is equal to the sum 
of the increments from that state due to the increment of 
each variable separately. 

It is important to notice that, while a uniformly varying 
function of two variables has a constant rate with respect 
to each variable alone, it has no fixed total rate with respect 
to both variables changing simultaneously. In the case 
illustrated, 

OT increment of J/iV^ ^,^_ 

^— = —j=^=^ = tan QMT. 

is the corresponding total rate of MJV, but in general any 
change in the relative value of JVJ? and J^S will cause a 
change in the total rate. Thus as the ratio NR/RS 
changes through all possible values, 0, the angle which the 
vertical plane MNS makes with the plane ZX changes, 
and the right line MT, cut from the given plane by the 
plane MNS revolving about MN as an axis, will in succes- 
sion coincide with all right lines in the given plane which 
pass through M. Hence, depending upon the ratio of in- 
crements of the variables, the total rate of any uniformly 
varying function of two variables with respect to both 
variables changing simultaneously, may have any value 
from zero to the tangent of the angle made by the graphic 
plane of the function with XK, the numerical value of 
which is called the slope of the plane. 

All functions of two variables not of the first degree with 
respect to each variable do not vary uniformly with respect 
to both variables, and their graphic surfaces are curved. 



SB DIFFERENTIAL CALCULUS, 

66. The Calculus is that branch of mathematics in which 
measurements, relations, and properties of functions and 
their states are determined from their rates of change. It 
is generally separated into two parts, called, respectively, 
Differential and Integral Calculus. 

Differential Calculus embraces the deductions and ap- 
plications of the rates of functions. 



DIFFERENTIAL CALCULUS. 



PART I. 

DIFFERENTIALS AND DIFFERENTIATION. 



CHAPTER IV. 
THE DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT. 

FUNCTIONS OF A SINGLE VARIABLE. 

67. An arbitrary amount of change assumed for the in- 
dependent variable is called the differential of the variable. 

It is represented by writing the letter d before the symbol 
for the variable ; thus dx^ read " differential of ^," denotes 
the differential of x. 

It is always assumed as positive, and remains constant 
throughout the same discussion unless otherwise stated. 

68. The differential of a function of a single variable is 
the change that the function would undergo from any state^ 
were it to retain its rate at that state, while the variable 
changed by its differential. 

The differential of a function is denoted by writing the 
letter d before the function or its symbol. 

59 



6o DIFFERENTIAL CALCULUS. 

Thus, d2ax^^ read " differential of 2^;\;V' ii^dicates the 
differential of the function 2ax^. 

Having J* = log ^ ax^^ we write dy =^ d log Vax^. 

~~dx denotes the differential of j; regarded as a function 
dx 

of X ; and —dy is a symbol for the differential of the in- 

dy 

verse function ; that is, of x regarded as a function of y. 

The differential of a function which varies uniformly 
with its variable is equal to the change in the function cor- 
responding to that assumed for the 
variable, because its rate is con- 
stant. 

Thus, let PA be any state of the 
uniformly varying function whose 
graph is the right line AB. As- 
sume J^J^ =dx. Then QB = dy, the corresponding change 
in the function, is the differential of the function. 

The differential of a function which does not vary uni- 
formly with its variable is not, in general, equal to the cor- 
responding change in the function, because its rate varies ; 
but it is equal to the corresponding change of a function 
having a constant rate equal to that of the given function 
at the state considered ; or, in other words, it is the change 
that the function would undergo were it to continue to 
change from any state, as it is changing at that state, uni- 
formly with a change in the variable equal to its differen- 
tial. 

Thus, let PA be any state of a given function whose 
graph is the curve AM. Assume PR = dx. 

QM is the corresponding change in the function ; but 
QB, the corresponding change in the function represented 




DIFFERENTIAL— Differential coefficient. 6i 




by the ordinate of the right line AB drawn tangent to 
AM at A, is the differential of 
the given function correspond- 
ing to the state PA. The func- 
tion whose graph is AB has a 
constant rate equal to that of the 
given function at PA., and QB is 
the change that the given func- 
tion would undergo were it to 
continue to change from the state PA., as it is changing at 
that state., uniformly with a change in x equal to dx. 

The differential of a function which does not vary uni- 
formly with its variable may 
be less than the correspond- 
ing change in the function. 
Thus, QB, < QM, is the dif- 
ferential of the function rep- 

^^ Q resented by the ordinate of 

the curve AM, correspond- 
— ^ ing to PA. 
A train of cars in motion affords a familiar example of a differen- 
tial of a function. 

A a; B CD E 

> 

Suppose that a train of cars starts from the station A, and moves 
in the direction AE with a continuouslv increasing speed. Let x de- 
note the variable distance of the train from A at any instant ; it will 
be a function of the time, represented by t, during which the train has 
moved, giving x — f{i). 

Suppose the train to have arrived at B, for which point x = AB. 
Let BD represent the distance that the train will actually run in the 
next unit of time, say one second, with its rate constantly increasing. 

Let BC represent the distance that the train would run it it were 
to move from B with its rate at that point unchanged, in a second. 




62 DIFFERENTIAL CALCULUS. 

Then will the distance ^C represent the differential of x regarded as 
a function of /, corresponding to the state x — AB ; and one second 
will be the differential of the variable. 

69. From the definition of a differential of a function, 
and from § 59, it follows that a differential of a function is 
the product of two factors, one of which is the i-ate of 
change of the function at the state considered, and the other 
is the assumed differential of the variable. Hence, the dif- 
ferential of any given function may be determined by find- 
ing its rate, by the general rule, §62, and multiplying it by 
the differential of the variable. Thus, having the function 
2x'^, we find 



limit 



z(x^h) 



hy - 2x'~\ 

\:=^x=^ rate corresp. to any state. 

^xdx is, therefore, a general expression for the differen- 
tial of 2jr^, and is written d 2x^ = ^xdx. 

Its value corresponding to any particular state is obtained 
by substituting the value of the variable corresponding to 
the state ; thus, for :r = 2, we have {d 2J\:^)^=2 — ^dx. 

70. Since, in the expression for the differential of a func- 
tion, the rate of change of the function is the coefficient of 
the differential of the variable, it is, in general, called the 
" differential coefficient^'' and may be determined by the 
general rule, § 62. 

The differential of a function is therefore equal to the 
product of the differential coefficient by the differential of 
the variable. 

It follows that the differential coefficient is the quotient 
of the differential of the function by the differential of the 
variable. Thus, having d 2x^ = /^xdx., 4X is the diff;. rential 
coefficient. In general, having y^=f{x), and representing 



DIFFERENTIAL— DIFFERENTIAL COEFFICIENT, (^l 

its differential by dy or —dx^ its differential coefficient is 

dy/dx. 

The expressions {dy/dx)^^'^ and dy' /dx' are used to de- 
note the particular value of dy/dx corresponding to x = x' 
and 7 = j'. 

Thus, having jF = 2^^, then dy/dx = 4X, and (<^/^x)(2j = 8. 

Having _y ==/(-^), in which jv is any function of any vari- 
able X., let y' denote the new state of the function corre- 
sponding to any increment of the variable, as h or /\x^ and 
let A J = _y' — jj^ represent the corresponding increment of 
y. Then (§ 62) 

limit /(^ + ^) - /(•^ ) ^ j-^^/jZJ' 3^ limit Aj ^ 4^ 
/^;^^o /; h Ax:^^o/xx dx' 

Since the increment of the variable, represented by h or 
A^, varies, it may happen that h = A^ = dx. It is exceed- 
ingly important to observe, however, that the correspond- 
ing value oi y' — y or l\y is 7iot, in general, equal to dy ; 
for that would give 

\ h Ih^dx \^xl ^:c=.dx dx' 

which, in general, is impossible, since dy/dx is not a value 
of the ratio {y' — y)/h, but is its limit under the law that 
h vanishes. 

If, however, the function changes uniformly with respect 
to the variable, {y' — y)/h will be constant for all values of 
^^ (§59)) and^' —y will be equal to dy when h is equal to 
dx. 

71* The following are important facts in regard to a dif- 
ferential coefficient : 



64 



D IFFEREN TIA L CAL CUL US. 



It is zero for a constant quantity. In other words, a 
constant has no differential coefficient. 

It is constant for any function which varies uniformly. 

It varies from state to state for any function which does 
not vary uniformly. 

In general, therefore, it is a function of the variable, and 
has a differential. 

It is positive for an increasing function, and negative for 
a decreasing one. § d^t' 

It may have values from — oo to + <^' 

Having represented a function by the ordinate of a curve, 
the differential coefficient is equal to the tangent of the 
angle made with the axis of abscissas, by a tangent to the 
curve at the point corresponding to the state considered. 
§62. 

Thus, assuming PR = dx, the differential coefficient of 
the function whose graph is the curve AM, at the state 
PA, is 

dy/dx = tan X£A — tan QAB. 

It should be noticed that dy/dx is independent of the 











Mi 








iY 


A 


^ 


^ 


I 
dy 










Q 


E^ 


^^^ 





P 


dx 


R 


r' X 



value assumed for the differential of the variable ; for if 
PP' = dx, then Q'D = dy, and we have, as before, dy/dx = 
tan XEA. 

In this illustration the function is an increasing one, its 




DIFFERENTIAL— D IFFERENTIAL COEFFICIENT. 05 

differential coefficient is positive, and the angle XEA is 
acute. § 63. 

In case the function represented by the ordinate of 
AM is a decreasing one, its 
differential coefficient corre- 
sponding to PA is negative, 
and the angle XEA is then 
obtuse. § d^' 

If for any value of the 
variable the differential coefficient is zero, the function is 
neither increasing nor decreasing, and the tangents at the 

corresponding points of the 
graph of the function are 
parallel to the axis of x. 

If the differential coeffi- 
cient is infinite, the rate of 
the function is infinite ; and 
the tangents at the corresponding points of the graph of 
the function are perpendicular to the axis of x. § 63. 

If for a finite value of the variable the state of a function 
is unlimited, its corresponding differential coefficient will 
also be unlimited. Thus, having ji; =/(^) and/(^) = co, 

limit 



' 


A 




A" 




^ y 




^^/ 


y 




p y 


P' 


P" X 



(^yUx). - '^Zl [(/{a + h) -Aa))/h\ 



Hence, 



(dy/dx). + e = [/(a + h) -f{d)\/h. 



in which e vanishes with h. f{a -\-}i) is not, in general, un- 
limited, therefore {dy/dx)a = 00 — ^ = 00 . 

The principle is not necessarily true for an infinite state 
corresponding to an infinite value of the variable, for in 
that case /(a -|- A) will also be unlimited. 



66 DIFFERENTIAL CALCULUS. 

72. The following facts concerning a differential of a 
function should now be apparent : 

It is zero for a constant. 

It is constant for any function which varies uniformly. 

It is a function of the variable for any function which 
does not vary uniformly ; and in such cases it has a differ- 
ential. 

Its value depends upon that of the differential coefficient 
and that assumed for the differential of the variable. 

It may have values from — 00 to -I- 00. 

It will be numerically greater or less than the differential 
coefficient depending upon whether the differential of the 
variable is assumed greater or less than unity. 

It has the same sign as its differential coefficient, and 
therefore is positive for an increasing function and negative 
for a decreasing one. 

Functions which are equal in all their successive states 
have their corresponding differentials equal. 

73 • The differential coefficient of aiiy function is equal to 
the reciprocal of the corresponding differential coefficient of its 
inverse function. 

Let J =/(^) . . . (i) and x = F(y) ... (2) be any direct 
and inverse functions. Let L^x and l^y represent, respect- 
ively, any set of corresponding increments of x andjv in (i). 
It follows (§ 4) that they will represent a set of the same in 
(2), and we have 

Ay/ l\x = i/{/\x/ /\y). 
Hence (§ 70) 

^y_ ^ limit A^ = lini —- = —1—. 

dx Ax-m^o /\x l\x/ l\y dx/dy 

To illustrate, let the function y be represented by the 



DIFFERENTIAL— DIFFERENTIAL COEFFICIENT, 6^ 



ordinate of the curve AM. Assume dx=^FR^ and from 
the figure we have, corre- 
sponding to the state PA^ 

QB dy ^ ,^ 

The inverse function will 
be represented by the ab- 
scissa of the curve AM re- 
garded as a function of the 
ordinate; and assuming dy 
= J^Z, we have for the state XA, corresponding to A, 

HE/ AH = dx/dy ^ tan EAH, EAH = 90° - QAB, 
Hence, 



L H dx ^/ 


3 


dy 
Y 


A 


>/ M 


dy 



X 




/ 



P da: R 



tan QAB = cot EAH = 



tan EAH' 



or 



dy 
dx 



dx/dy 



It should be observed that, in general, dy in the first 
member of the above equation is not the same as dy in the 
second; for the first is the differential of ^j' as a function, 
and the second is a differential of y as the independent 
variable. The same remarks apply to dx^ in the two menii- 
bers, taken in reverse order. 

The figure illustrates the differences referred to. 

74* The differential of the su?n of any finite nur?iber of func- 
tions is equal to the sum of their differentials. 

Let y^^v — s-\-w-\- etc., in which v, s, w, etc., are func- 
tions of any variable, as x. Increasing x hy ^x, we have 

y + l\y = 27+ Az;— (s -\- /\s) -\- w -^ Aw -\- etc. 
Whence 



Ay 



= Av — As -{- Aw -{- etc., 



68 



DIFFERENTIAL CALCULUS. 



and (§ 70) 

dy 



limit \ ^_ 



Isw , ^ ~\ dv ds , dw , 



dx 



As 

_ Ax Ax 

Therefore 

d{v — s + w + etc.) = dv — ds 4- dw -f etc. 

It follows that ^/le differential of the sum of any finite 
number of functions and constants is equal to the differential 
of the sutn of the functions. Thus, C being constant, 

d[f(x) + C]= df(x). 

If corresponding differentials are equal it does not follow 
that the functions from which they were derived are equal, 

75* T^^ differential of the product of any number of func- 
tions is equal to the sum of the products of the differential of 
each function by all of the other functions. 

Let V = yz be the product of any two functions of any- 
variable, as X, then 

V -{- Av = {y -\- Ay) {z-\- A z) =yz-\-z . Ay-\-y . A z-\- Ay, A 2, 

whence Av ^ z . Ay -\- y . Az -\r Ay. Az \ and (§ 70) 



— — ^^^^^^^ - — - — hm z -^^ + V h A V 

dx A^^->o ajc LAj*:"^AJt: "^ 



= zdy/dx -\- ydz/dx. 

Therefore, dyz = zdy + ydz. 

To illustrate, let ONPM be a state of a rectangle with a 
variable diagonal represented by 
X. Two adjacent sides, denoted 
by z and y respectively, will be 
functions of x^ and yz will be the 
variable area of the rectangle. 
Assume dx^^PR^dc^di complete the rectangles PT'^jf^s 



dz 
V 


^dy 




z 

S 



DIFFERENTIAL— DIFFERENTIAL COEFFICIENT. 69 

and FS=zdy. Then, since dyz = ydz -\- zdy^ we have 
^ (rect. OF) = rect. FT-{- rect. FS, which is the amount 
of change required by the definition of a differential. § 68. 
It follows that t^e differential of the product of a function 
and a constant is equal to the product of the cotistant and the 
differential of the function. Thus, C being constant, 

dCf(x) = Cdf(x). 

Let vsu be the product of any three functions of the same 
variable. Place vs =■ r, giving vsu = ru. 

Differentiating, we have dvsu =^ dru = rdu + udr^ in which 

dr = vds + sdv. Hence, by substitution. 

dvsu = vsdu + vuds + sudv. . . . (i) 

In a similar manner the principle may be established for 
any number of functions. 

Dividing each member of (i) by vsu^ we have 

dvsu _ du ds dv 

vsu ~ u ~^ s ' V* 

Similarly, it may be shown that the differential of the 
product of any nuinber of functions divided by their product 
is equal to the sum of the quotients of the differential of each 
function by the function itself. 

EXAMPLES. 

4(a -1- x){b + x)\ = {b-^x) d{a^ X) -f {a-\- x) d{b + x) = (a + b-^2x)dx. 

d[2,{c — ^)J = — 2dx. d[(a -\- x)x] = xd(a -\-x)-\-ia^ x)dx. 

d \{a -f- x)x'\ _ d{a -\- x^ dx _ dx _, ^ 
(a -\- x)x a -\- X X a -{- x x ' 

76. The differe7itial of a quotient of two functions is equal 
to the denominator into the differential of the manerator^ 



70 DIFFERENTIAL CALCULUS. 

minus the numerator into the differential of the denominator^ 
divided by the square of the denominator. 

Let y = v/s^ in which v and s are functions of any varia- 
ble. Then v — sy, and 

dv = sdy + yds = (s'^dy -|- vds)/Sj 

whence dy — (sdv — vds)/sl 
C being a constant, we have 

d{C/s) = - Cds/s^ and d(v/C) = dv/C. 

EXAMPLES. 
d [jc/ii + x)] = dx/{i + x)\ dis/x) = - 3dx/x^. 
d\x{x + i)/(x - I)] = {pc" -2.x- \)dx/{x - if. 
d{x/3) = dx/3. d{2x/sa) =2dx/Sa. 

77* ^^^ differential coefficient of y regarded as a function 
of X is equal to the product of the differential coefficient of y 
regarded as a functioji of u, by the differential coefficient of u 
regarded as a function of x. 

Having jv =/( 2^), and u = 0(.^), let A^, Hu and t^y 
be corresponding increments of x., u and y, respectively. 
Then (§ 4) A u is the same in both cases, and 

Aj/A^ = ( A;v/A?^) X ( A2//A:r). Hence, 
aJ.'^o V^yl ^^~\ = lini [aVA2/] X lim [A^A^], 
and (§70) dy/dx = {dy/du) X {du/dx). 

Similarly, having ji^ z= f(ji)^ u = (p{x), x = i^is), we find 
dy/ds = (dy/du) X (du/dx) X (dx/ds) ; 

and the same form holds true whatever be the number of 
the intermediate functions. 



DIFFERENTIAL— DIFFERENTIAL COEFFICIENT. 7 1 

Having y =f{u), and x = ^'{u), we may write u = <p{x), 
and dy/dx^^ {dy/dii) X {du/dx), but (§ 73) du/dx^= i/{dx/di/). 
Hence, 

dy/dx = {dy/du)/{dx/du). 

That is, ^/z^ differential coefficient of y regarded as a func- 
tion of X is equal to the quotient of the differential coefficient 
of y regarded as a function of u, by the differential coefficient 
of X regarded as a function of u. 



EXAMPLES, 



Given 

y = au^, u = bx. . 



y = f{^u\ X = (p{u), X = ^{s). 
y — u^, X = 3u, X = 2/. . 
y = f{ii), u — F{s), z = 1p{s). 



dy/dx = 2a 6^ X. 

dz/dx = 2ap. 

dy _ dy/du dx 
ds dx /du ds ' 

dy/ds = i6s^/g. 

dy du/ds dy 
dz dz/ds du ' 



y ■= f{u), V — 0{u), V — ip{s), z = F{s), z = Fi{x) 



7. y=f{s)=f{x^h),%^ 
dy _ dy 



dy^ __ dy/du dv/ds dz 
dx ~ dv/du ^ dz/ds ^ d^' 

.'. s = x-}-A. 
dy dy 



Then i'^- _ ^ X A and ^ - ^ ^ _± 

dx ds ^ dx' dh ~ ds ^ dk 

Pj ds d> dv dv dy 

But — = — = I, hence -- =r - = — . 

dx d/i ax dh ds 



^2 DIFFERENTIAL CALCULUS. 



CHAPTER V. 
DIFFERENTIATION OF FUNCTIONS. 

FUNCTIONS OF A SINGLE VARIABLE. 

78. The differential of any function of a single variable 
may be determined by applying the general rule, § 70, § 62, 
and multiplying the result by the differential of the variable; 
but by applying the general rule, § 70, § 62, to a general 
representative of any particular kind of function, there will 
result a particular form, or rule, for differentiating such 
functions, which is generally used in practice. 

79. The differential of any power of any function with a 
constant exponent is equal to the product of the exponent of the 
power, the function with its exponent diminished by unity, and 
the differential of the function. 

Let J/ = x^, in which x is any variable and n is any 
constant. Then, increasing x by h, we have j[§ 70) 



dy __ limit 



Hx + hY - ^" "I 



Placing X -{- h = s, whence h = s — x, and as ^ b-> o, 
s B-> X, we have (§ 40) 

^ ^ limit p"-^n = ^^n-1 ^^^ dx" = nx"-idx. 

Having y'^, in which y is any function of any variable, 
as xy, we have (§ 77) 

dy^dx = {dy^'/dy) X (dy/dx). 



DIFFERENTIA HON OF FUNCTIONS. 73 

Hence, 

dy-^/dx = nf-^(dyldx), and tT ^ ny^-^dy. (i) 
Substituting \/n for « in (i), we have 

dyV« = -y/'*-!^ = ^-y'^dy -= dy/n |/y^^. 

Hence, //^^ differential of the n^^ root of any function is 
equal to the differential of the function divided by n times the 
n^^ root of the n — 1 power of the function. 

EXAMPLES. 

1. ^ V^ = dx/2. VJ. 12. a' f ^ = dx/l ^7K 

2. dx^ = 2xdx. 13. dx^^ = —nx-^-^dx. 

3. dx^ = 3x^dx. 14. </jf-* = — 4x-^dx. 

4. ^/4x* = i6x^dx, 15. aT^r"^ = — —x'^dx. 

2 

5. ^a^2 _ 2axdx. 16. </;«:" »^= - -;p ^~ dx. 

n 

6. </(log xY = 2 log X d\og X. 17. </(a'»)2 = 2a^da^. 

7. ^ sin^ ;t = 3 sin'^ x d sin jf. 18. ^ tan« ;r = w tan»-i ;t i/ tan x. 

8. 0^3;^^ _ g^VAT. 19. fl'[3(fl + a:2)3] = i8(« + Ar2)2;r^;p. 

9. dl^^Tlx^/'i) = 4;rxVjc. 20. d{27tax'^ / z) = ^iiaxdxji. 

,^ ^ "/ — T";. / I \~n~v / 21. di^xY^ = 20(2xydx. 

II. ^ V;(;3_^3 _ 2x^^x/2 Vx^—a\ 22. ^[2x '^ /7] = 3^x/i4;t ^ . 

23. dia-^x^y = 3(a + x2)2 ^(a+;«r2) = sia+x^ 2xdx = 6x{a-\-xydx. 



24. d^a-{-x^ = d{a + ;*:2)/3 f (a + x'^f = 2xdx/3 ^{a -\- x^f. 

25. d{2xy = 2{2x)d{2x) = 8xdx. 



74 DIFFERENTIAL CALCULUS. 

26. d^Zx'f - 2{2x')d{2x') - Idx^dx. 

27. d{ax^f = 2(ax^fd{ax^) = 6a^x^dx. 

28. d{3x)~^ = - 2(3xy^d{3x) = - 6(3xf^dx. 

29. dia^- x-'f = l{a^ - x^y^dia-" - x^) = - (a'^ — x^)~^xdXo 

30. d[{x^ -\- a)(3x^ + b)] = (15^' + 3^^' + 6ax)dx. 

31. d[xy V{i - xy] = 3x'dx/{i - x^)"\ 

32. d{a -\- X — 3x^ 4" 4^^] = (i — 6x + I2x^)^x, 

33. ^[(i + x2)(i — x^)] = (2x — 3X'' — t^x^)dx. 

34. d{a -|- bx^Y = bmn{a -f ^;c'")«-ia:»»-i^^. 



35. d\{2x'' - i)/x Vi -\-x'] = {4x'^-\-i)dx/x'^ii-\-x''yi\ 

36. d[x / Va' - x'] = a'^dx/ia'' - x^", 

37. ^(l + ^H= - dx/2{l + X)'. 



38. d\/a'-\-x^ = xdx/ Va'^-\-x^. 

40, ^[;t:"/(i + ;»:)"] = nx'^-'^dx/{\ + ;t;)w+l. 



41. d\i/ Vl - ;^2] = xdx/{i - ^2)3/2. 



42. ^[x/ Vi-x] = {2- x)dx/2(l - xf'K 



43. d[x/ Vl - x^ = dx/(i - xy^. 

44. dlx^/{i - x^Y'"'] = 3x''dx/{i - xyi\ 



45. 4 Vl -^x/ Vi-x'\=^ dx/{i -x)Vi- xK 

46. d{x'^^/{i +;c2)«] = 2nx'i^--^dx/{i +;t:2)«+l. 



47. 4(1 - x)/ f I + x'^j = - (I + xW(i + ^y""' 



48. ^[x/(x - i^I - ^2)] := _ ^;j:/ |/i _ jc\x- Vl - x'% 



49. dVx'^/{2a — x) = {3a — x) Vxdx/{2a — x)^ 



50. d{- Vd^ - x^/d^x) — dx/x Vd^ - x2 



51. d[ax/{x -{-Va-\- x')] = a'dx/{x -{- Va -{- x'f Va + x'^. 



DIFFERENTIA TION OF FUNCTIONS. 75 



52. d{x/a' \'x^ + a") = dx/ ^{x' + d^f. 

53. ^[x/(i +x)]w — ^jr"-Vjr/(i + jc)"+i. 

54. ^[(« + bx^l'')/c- ^x'] = il;x^l^-sa)dx/4c i/I^. 

55. d[x^/{a -\-x^y] = 2x(« - 2x^)dx/{a + ^3)3. 



56. dxid" + x«) l^a-^ - x^ = («4 + aV2 - 4x4)^x/{ /^ - x^)'/'. 

57. 4x^/='(i + xyi\/{i - x)^/^j = (3+2X - 3x2) v^^;^/2(i _ xyi\ 



58. «'(i -|- '^) '^^i — -^ = (i — 3x)dx/2 Vi — X. 



59. ^[ Vr - Vx/ Vi -^Vx]=- dx/i{\ + Vx) 4/^r^^^. 



60. ^ i^x + vFT^ ^ dx s/x^ Vi -f ;rV2 vr+^. 

61. M = [a — V^^H- (^' - ^')'"]'/''. 

1°. d{a-d/ Vr+(^2_^2)2/3^ ^ |-^/2x V^-{-2{c''-x'')-H-2x)/3]dx 
= \bl2x Vx- 4x/3{c' - xy]dx. 
Hence, du = ^-[a - 4+(.^- x^'^l -^[ -±-_ ^-^^—Idx 

^[_ Vx J \_2xVx 3(^2-.v^)i/3j 

62. In the parabola y^ = gx, find the rate of y with respect to x 
when X = 4. What value will x have when rate of y equals that of jf ? 
When rate of y is the greatest ? When the least? 

dy/dx = g/2y = ±3/2 V x. .-. (dy/dx)x=4 = ± 3/4. 

dy/dx = ± 3/2 \/x = I gives x = 9/4. 

dy/dx is the greatest when x = o, and the least when x = cc, 

63. Find the slope of the curve y = ± \/g — x^ when x = 2. Find 
values of x and ^ when the slope is i. 

Ans. ± 0.894. x' = ± 1/4.5, y = :F 4/4.5. 

64. Find the angle which the curve y = x/(i -\- x'^) makes with X at 
their point of intersection. Ans. 7t/4. 

65 Find the angle at which the curves^ = lox and x^ -{-y^ = 144 
intersect. Ans. 71° o' 58". 



^6 DIFFERENTIAL CALCULUS. 

66. Find the value of x at the point where the slope oi y^ — /^x^ is 
unity. Ans. i/g. 

67. At what rate does the volume of a cube change with respect to 
the length of an edge ? Ans. 3 (edge)"^ 

68. Find the angle that a tangent to the curve x^ = 6y^ -\-3y-\- i, 
at the point (8, 3), makes with the axis X. Ans. tan-'(i6/39). 

69. Find the rate of change of ( \^x -j- 3/ax'^) when x = 3. 

Ans. 1/2I/3 — 2/ga. 

70. Find the rate of change of the ordinate of a circle with respect 
to the abscissa. Ans. "^x/ ^ R'' — x'^. 

71. Find the rate of change of the ordinate of an ellipse with re- 



spect to the abscissa. Ans. T bx/a \/a^ — x^. 

80. Tke differential of the logarithm of any function is 
equal to the differential of the function divided by the function. 
Let jj' = log X, then (§ 70, § 42) 

x-\-h 
dy_ ^ limit log(^ ^-h)- log x _ ^^ x 

dx h-m^o h h 

= lim , = Iim ; 



= - lim log ( I + -)^ = — log <? = — . 
X ^ \ ' xJ X ^ X 

Hence, d log x = dx/x. 

Since loga e = Ma , it follows that d loga x = Ma dx/x. 

Having >' =f(x), we write (§77) 



DIFFERENTIA TION OF FUNCTIONS. 77 

d\ogy __ dXogy^ ^ — L x -^ 

dx . dy dx y dx 

Hence, d log y = dy/y. 



EXAMPLES. 

I. flTlog ^ = {de^)/e''. 2. ^ log sin jf = (aT sin x)/sm x, 

3. d log x'^ = dx'^/x^ = 2xdx/x^ = 2dx/x. 

4. </log |/^= d^x l^~x = (ar;«r/2|/7)/|/J=^V2-^« 

5. d log jf"' = dx'^/x^ = nx^-'^dx/x^ = ndx/x. 

6. d log i/x2 - 1 = d |/x2 -iZ-l/jt^-i = ;>:^;<:/(;c2 — I). 

7. ariog[(i +^)/(i - x)] =4(1 + xj/{i - x)]/[(l + ^)/(l - AT)] 

=z2dx/{i — x"^). 

8. </[log log x"] =d log Ar/log jf = (^j;/;c)/log x = dx/x log x. 

9. fi?[Ar log jr] = jroT log ^c ■\- log JCdfx = (i -|- log x)dx. 

10. flT log (x2 + X^) = d(x^ + X^ix' -t- X3) = (2X + 3X^)dx/(x^ -f ^3). 

11. ^(log x)^ = m(\og x)^-'^ dlog X = m{\og x)^-^dx/x 

12. d(i/log x'*) = — d log ;t«/(log x'^f = — ndx/x(\og x^)\ 

13. d log |/(i + x)/(i - X) = dx/{i - ^2). 

14. ^log [(I +;t;V(i - ^^)] = 2dx/3x^i-xh 

15. /^"» (log xY = [w(log x)« + «(log xY-^x^-^dx. 

16. a' log [y^3qr-i/ ^^z _ I] = _ 2,x''dx/{x^ - I). 

17. dT log [(I + |/i^^2)/^-j ^ _ ^^/^ 4/1 _ ;p«. 

18. ^log [(I + |/^/(i - f'^)] = c/V(i - ^) 4/^. 

19. a' log [x 4/^ + Vr^^"'] = - dx/X^x'^ ^ I. 

- , i/;*;' -I- I — I 2^;k: 

20. ^log -^-^==3ir = . 

V;c« + I -f- I ^ 1/^2 _|_ J 



yS DIFFERENTIAL CALCULUS^ 



i/x"^ -\- a- — X — 2dx 
21. d log ' — 



22. ^ log 



23. a' log 



^ \ -\- x -\- \/\ — X _ — dx 
\/i-^x — \/i - X ^\/'i~ 



a/Vi + x^±x_ ^ d^ 



Vi + x^ —X Vi + ^ 

24. d log ^'^-'^i = _ xdxjia} — x\ 



25, d log |/«2 _|_ ^-2 :^ x^x/0?-^ -f- X""). 

26. ^ (log [(a - ;tr)/(a + ^)]/2«) = dx/{x'^ - flS). 



27. ^nog (x/|/i +x'^) = ^x/x(i +x^). 



28. a' log (x± |/x2 ±ii^) — ±dxl \/x'^ ± a^. 

29. «'[log log... (repeated n times) ofjr] = i/r/xlogx(log)'x...(log)'^-'x. 

30. d log [log {a + bx'^)'\ ~ nbx''^-^dx/{a -\- dx^) log {a -\- dx^). 

31. zi = (log ;*:«)"*. Put log x^ = jj/, then (§ 77) 

^z< = d{log x'^)'^ = 7ny'"^-\n/x)dx = f/in(\og x'')^-\dx/x. 

32. Find the rate of change of a logarithm in the common system 
with respect to the number. Ans. J/10/number. 

33. Find the slope of the curve y = \oga x at the point (i, o). 

Ans. Afa. 

81. T/ie diffei'etitial of any exponential function with a 
consta7it base is equal to the product of the function^ the loga- 
rithm of the base, a?id the differential of the exponent. 

Let y ^ < then (§ 70, § 43) 

dy limit ^*'^'' — ^"^ r 1- a^ — \ 

dx h-mH>o h h ^ 

Hence, da^ — a^ log adx. 



DIFFERENTIATION OF FUNCTIONS. Jg 

Having a^, in which j- = /(x), we have (§ 77) 

da^ _ day ^V _ y ^ ^y 

dx dy dx dx ' 

Therefore da^ = a^ log ady. 

It follows that d^y = eydy. 

EXAMPLES. 

1. da^"^ = a^^ logadx^ = 2a*' Jt: logadx, 

2. c/J""^ ^ = a^°^ ^ log a ^ log X = a^'^S ^ log a dx/x, 

3. da^^ = a"^ \oga d ^'x — a ^^ log a dx/2 Vx, 

4. da ^ — a ^ \ogad{l/x) = — a ^ [ogadx/x"^. 

5. da^'x'^ = a^dx-^ + x'^da^ — a^x'^-\x log a + «)^. 

1 _ 1 

6. de~ "" = e ^dx/x"". 

7.^ a'[(^^^ — e-x)/2\ = (^* + e-^)dx/2. 

X _X X X 

8. ^[a(^" + ^ «)/2] = (.'« - e'"^)dx/l, 

9. ^i?* log X = (i/x -f- log x)e^dx. 

10. «'[(^* - \)/{e^- + I)] = le^'dxl^e^ + i)^ - 

11. a' log \[e^ - i)/{e^ 4- i)] = 2e''dx/(e^^ — l), 

12. ^" log (^* 4- ^-a-) = (^a^ _ e-'')dx/(e''-]-e-^). 

13. a' log {e^ — e-^) = {e^ + e-^)dx/{e^ - e-% 

14. 4(a* — i)/(a* 4- i)] = 2a* log a a'x/(o* -1- l)«. 

15. a'(a* + xf = 2(a* + x)(aa; log a -\- \)dx. 

16. a^^*(i — A^) = ^*(i — 3^2 — x3y^_ 

17. flt[(^ - e-^)/{e^ + ^-^0] = 4^V(^* + ^"^)*- 

18. a[x/{e=' - i)] = [^^(i ~ x) - \\dx/{e^ - 1)2. 

19. dx'^{\ + xY — nx'^-\\ + x)"-i(i + ■zx)dx. 



8o DIFFERENTIAL CALCULUS. 

20. When jr — o, find the inclination of the curve jv = lo'" to X. 

Ans. 66° 31' 30". 

82. Logarithmic Differentiation. — The differentiation 
of an exponential function, or one involving a product or 
quotient, is frequently simplified by first taking the Na- 
pierian logarithm of the function. 

Thus, let u = f^xvi which y and z are functions of the 
same variable. 

.•. log u=^ zXogy 

and (§ 80, § 75) 

du/u = zdy/y -f- log ydz. 
Hence, du = dy'' = zy^-^dy -{- y"" log ydz^ 

which is the sum of the differentials obtained by applvinq 
first the rule in § 79, then that in § 81. 

EXAMPLES. 



i/{x 1)' log « = I log (x - I) 



V{x - 2f V{x - 3)' 

- I log (^ - 2) - I log {x - 3), 
du ^ dx 3 </x 7 dx 'jx^ -\- 30X — 97 



u 2x— I /^x—2 3x—2> i2{x-i){x—2){x—2) 

du=- (^-1)^7^; +30^ -97)^^ 

2. dx" = jr*(i + log x)dx. 

I I-2JC 

3. dx'^ — X -^ (i — log x)dx. 

4. dx^ = x^ X (log' X -f- log X -f- l/x)dx. 



dx, 



5. dx\/i — x{i 4- x) — (2 -j-x — <sx'^)dx/2\/i — X. 

{x-if^^ ^ _ (x-i)^/'(7x'-j-3ox-g7)dx 

(;, _ 2)3/4(^ - 3)'/' i2{x-2y^'\x-3y°^^ ' 



DIFFERENTIA TION OF FUNCTIONS. 8 1 

7. d{x/ny^ = «(V«)"^Li + log {x/n)]c/x. 

8. ^x^/^ = x^^"" \og{e/x)dx/x\ 

9. de^"^ = e'^^e^dx. - 10. dx^ = x«V^;tr(i + x log x)/x. 

11. d'-f'"^ = ^»^ji;*(l -\- log jr)i/^. 

1-2X 

12. dix'^ + a;^/*) = [;c* log ex — x '^ \og{x/e)]dx. 

13. M = :«■'<•» '». Put log X = z, then 

</« = ^;t;2 — (logx ^loga;-! _|_ ^^logj; [og jc/xy^tr = 2;ci°g^-i logxdfj;, 
a'(Iog ;t:)^ — [(log ;c)*-i + (log x)"^ log log x]^;c. 



14. d 



i/2.a:(r - x')3/-^ _ (- 8 ;c' + 24;^:^ - Jtr - 6K;t: 
(j, _ 2)^/3 ~ 3(^)'/'(l - x^W\x - 2)5/3* 



Trigonometric Functions, 
83. d sin X =: COS X dx. 
Let ^ be the increment of x^ then (§ 70) 

■^sin^ _ limit 
dx hm^ 



it Fsin (:<[: + i^) — sin or\ 
^o L li J 

I 2 sin - cos [x A I 



cos^. 



Having sinj, in which j =f(x)^ we have (§ 77) 

^sinj' _ ^sin_y ^y _ dy 

dx dy dx dx' 

Hence, ^sin^ = cosjf^. 



82 



DIFFERENTIAL CALCULUS. 



Similarly, by applying the general rule, § 70, and the 
principle in § 77, the differential of any trigonometric func- 
tion may be determined ; but it is perhaps simpler to make 
use of the relations existing between the functions. 



d COS X = ^sin {jt/2 



x) — cos {7t/2 — x)d{7r/2 — x) 
= — sin X dx. 



d tan 'x.— d 



sm^ 



dx 



cos X 



COS' X 



sec' xdx =r (i -)- tan' x)dx. 



d cot X = ^ tan ( ;r/2 



x)= — dx/sin' X = — cosec' x dx 
= - (i 4- cot' x)dx. 



d sec X = ^(i/cos x) — sin x dx/cos" x = tan x sec x dx. 

d cosec X 1= ^sec {n/z — x) = — cotiL cosec xdx. 

d vers x = ^(i — cos x) = sin x dx. 

d covers x= ^vers (rr/i — x) — — cos xdx. 

In order to illustrate the formulas for the differentials of the 

sine and cosine of any angle, 
let ACB = be any given 
angle. Assume BCN = d(p, 
and with any radius, as CO 
= R, describe an arc, as 
OMN. Then 

FM/R = sin <p, 

A_ CF/R = cos 0, arc MJV = Rd(p. 

The definition of a differential (^6;^), in this case, requires 
that sin and cos 0, retaining their rates at the states 
corresponding to = y4CB, shall continue to change from 
those states while increases by the angle BCJV = d(p. 




DIFFERENTIA TION OF FUNCTIONS. 



83 



Draw the tangent line to the arc at M^ and lay off MT 
equal to the arc MN = Rdcp. Through T draw TQ paral- 
lel to MP^ and through M draw MQ parallel to OC, 

Then QT a.nd — MQ are, respectively, the changes that 
the lines PM and CP would undergo were they to continue 
to change with the rates they have when 0= ACB, while 
<p increases by d<p. 

Hence, QT/R, and — MQ/R are, respectively, the 
changes that sin and cos <p would undergo under the 
same requirements. 

The angle M TQ — 0. Hence, 

QT=MT cos = ^cos (pd0, and QT/R=ds\n 0=cos (pd(p. 
—MQ=MT s'm 0=/? sin 0d(p, and MQ/R=d cos 0= — sin (pd(p. 

Similarly the formulas for the differentials of the othei 
trigonometric functions may be illustrated. 



Regarding the right lines 
PM, CP, OE, O'B, etc., as Q 
functions of the variable angle 
0, we have 




dPM=dR sin (})=R cos (pd0. 
dOE=dR tan (p---Rdcp /cos"^ (p. 
dCE — dR sec 

=i? tan sec (pd(p. 



dCF=dR cos (p= — R sin (pd(p. 
dO'B=dR cot (p^-Rd(p/s\n^ (p. 
dCB = dR cosec <p 

=—R cot cosec (pd<p. 



dOP = dR vers = /? sin 0^0. dO'Q=dR covers (p= — R cos 0^<?>. 



It is important to notice the difference between the dif- 
ferentials of the above lines, which depend upon the radius 



84 DIFFERENTIAL CALCULUS. 

of the circle used, and the differentials of the trigonometric 
functions which do not depe-^d upon any radius or circle. 



EXAMPLES. 

1. d sin x^ = ix ros x^ dx, 12. d cos x^ = — 2x sin x^ dx. 

2. d sin^ ;c = 2 sin X cos x dx. 13. d sin^ ^ = 3 sin* jt cos xdx. 

3. d zo^ X =-^izo^ xsinx dx. 14. ^cos^ jc = — 3 cos" j;sin ;f ^jf. 

4. ^ tan' j; = 2 tan jf dx/cos^ x. 1 5. </ tan^ ^ = 3 tan^ x dx/cos^ x. 

5. </cot' ;f = — 2 cot j;^;r/sin^ A-. 16. ^cot^^f =— 3 cot';r^jr/sin* at. 

6. d sec^ jr = 2 sec^ ;p tan x dx. ly. d sec^ ;r = 3 sec"^ x tan jra^x. 

7. ^cosec'^=— 2 cosec^ j;cot;r </x.i8. ^cosec^;c= — 3cosec^CDt;f ^;i;. 

8. d vers'-^ ^ =2 vers x sin ;if dfjc. 19. d vers^ jc =3 vers' sin x dx. 

9. d CO vers2x= — 2 covers x cos ^ ^^. 20. d cos ;r^ = — 3;^'' sin x^ dx. 

10. </ tan :r' = 2;i(r^;ir/cos'' Jf2. 21. </covers^;ir= — 3 covers'^ ^ cos ;«:</;»:. 

11. de^^^ ^ = r^^°^ ^ cos X dx. 22. d log cos x = — tan x dx, 
23. </ sin'* x^ = 2 sin ^V sin ^' = ^x sin a-" cos x^ dx» 



24. ^|/tan 2jr = sec' 2x dx/ |/tan 2x, 

25. d cos{i/x) = (i/^") sin (i/x)dx, 

26. d? tan" ;r' = 4^ tan jc' dx/cos^ x'*. 

27. </ cos mx = — sin wjf d(mx) = — m sin mxdx» 

28. ^ sin 2>x = cos 3;); ^ 3;^ = 3 cos "^x dx. 

29. </ sin' 2;i: = 4 sin 2x cos 2x dx. 

30. i/ sin^ajf = na sin^-^ajc cos ax dx, 

31. d tan"^ ^ = » tan"^~^ x dx/cos^ x, 

32. ^(tan X — j:) = tan' x dx. 

33. ^[(;r — sin X cos ^)/2] = sin' x dx. 

34. </ tan X sec ;f = (sec' x ■\- tan'* x) sec x dx, 

35. t/ tan a'"" = - a-"" sec' a"^ log a flJr/-^'^'. 



DIFFERENTIATION OF FUNCTIONS. 85 

36. ^[(i — tan j:)/sec x] = — (cos x -j- sin x)dx. 

37. ^[sin nx/cos'^ xl = « cos (« — i):r arx/cos"+^ ^. 



38. d tan Vi — X = — (sec r i — xydx/2 Vi — x. 



39. df T sin \/ X = cos Vxdx/4 Vx sin V;*:. 

40. d^(cos jf)^'° ^ = (cos jr)^'" ^ (cos X log COS X — sin' ;r/cos x)dx. 

41. d sin (sin ^) = cos x cos (sin jr)^jr. 

42. d COS (sin ;r) = — cos x sin (sin j:)d!vXr. 

43. 0^ sin (log nx) = cos (log nx)dx/x. 

44. ^;c^'" ^ = ^sin a; ^^^^ ^ log x + sin x/x)dx. 

(sin ajr)^_ a3 (sin ax)^~'^ cos (^;r — ax)dx 
^^' (cosJxY' ~ (cosl^f+1 • 

46. i/ sin^ :v COS X = sin'* x(3 — 4 sin" x)fl&P. 

47. d cos^ X sin' x = (2 cos'* x — s sin'* x) cos* x sin ;ir di'x. 

48. </[(sin X -|- cos x)/(sin x — cos j^)] = — 2dx/{s'm x — cos xy. 

49. flr(sin at)^^" ^ = (sin x)'^" ^ (i + sec'* x log sin ;«ry;c. 

50. Determine the manner in which the sine of an angle varies with 
the angle. 

The rate of change of sin x is (§ 70) d sin x/dx = cos x, from which 
we see that as (p increases from o to 7t/2, the rate is +» but diminish- 
ing; hence, the sine increases, but its increments decrease. 

From 7r/2 to tt the rate is — , and diminishing; hence, the sine di- 
minishes and its decrements increase numerically. 

From 7t to 37r/2 the rate is — , and increasing. That is, the sine 
decreases, but its decrements diminish numerically. 

From 37r/2 to 27r the rate is -{-, and increasing. That is, the sine 
increases, and its increments increase. 

In a similar manner the circumstances of change of each trigone^ 
metric function with respect to the angle may be determined, 

51. Assuming dx = 7t/4, we have, corresponding to :r = 7t/6, 

^ . V3 , ^ y ^ 

asiax^=—- — 7t. dcosx = ——. dtanx=—. 

8 83 



86 DIFFERENTIAL CALCULUS. 

d cot X =^ — 7t. d sec x = — . d cosec jc = —it. 

6 2 

52. Corresponding to x = 7r/4, we have 

a'sinx I ^cosx i ^ tan ;»: 



dx j^2 ^^ 4/2 ^^ 

d cot X _ d sec -^ _ /— d cosec Jic /— 

dx ' dx ' dx 

53. What is the value of x when tan x is increasing twice as fast 
as X? Ans. n/^. 

54. Find the rate of change of the tangent, regarded as a function 

of the sine of an angle. 

Since tan x =/r, and sin x = Fx, we have (^5 77) 

d tan x/dsin x—{d tan x/dx)/{d sin x/dx) 

■= (i/cos'^ ;r)/cosx = i/cos^^. 

In a similar manner the rate of change of any trigonometric func- 
tion regarded as a function of any other may be found. 

55. Find the slope of the curve y = sin x when x= o, x := 7r/4, 
X = Ti/i. Ans. I, |/i/2, o. 

56. When X = 7r/3, find the inclination of the curve y = tan x to X. 

Ans. 75° 57' 50". 

57. Find the angle which the curves^ = sin x and j = cos x make 
with each other at their point of intersection. Ans. tan"^ 2^2^ 

Inverse Trigonometric Functions, 
84. d sin-i X = dx/|/i - x^ 

Let = sin~^ x ; then Jt: = sin and dx/d(p = cos 0. 
Hence (§ 73), 

^sin~^ x/dx = i/cos = i/± r i — x'\ 

* The sign depends upon that of cos (p. Formulas involving the 
double sign in this article are generally written with the plus sign 
only, which corresponds to angles ending in the first quadrant. 



DIFFERENTIA TION OF FUNCTIONS. '^'J 

By applying the principle in § 77, it may be shown that 



d^vcT^ y = dy/Vi — y, in which j^ =/x. 



d cos-i X = d{n/2 — sin-^ x) = — dx/ 4/1 — x'. 
dtan-ix = dx/(i + x'). 

Let = tan"-^ x ; then 

X = tan 0, and dx/d(p = i + tan" <f>. 

Hence (§ 73), ^tan~^ x/dx = 1/(1 -|- jc*). 

d cot-i X — d(n/2 — tan-* x) = — dx/(l + x"), 



d sec-i X = dx/x Vx' — i. 

Let = sec~* x, 
then jc = sec 0, and dx/d(^ = sec tan 0. Hence (§ 73), 

^sec"* XI I I 



dx sec tan sec (pVsec'' — i xVx' — i 



d cosec-i X = d{7r/2 — sec~^ x) = — dx/xVx" — i. 



d vers"'^ x = dx/y 2X — x^ 

.et = v€ 
Hence (§ 73), 



Let = vers"* x ; then ;^ = vers 0, and -^ = sin 



I 



^^ sin |/i _ cos' 1^1 - (i - vers 0)' 

I I 



r 2 vers — vers'^ t 2x — j^^ 



88 



DIFFERENTIAL CALCULUS. 



d covers-^ x = d{n/2 — vers"^ x)= — dx/l/2x - x". 

Regarding as a function of 
the line PM^ denoted by% we 




. y 
E have = sin"^-^. Hence, 



d(t) 



4 



dy 






Similarly, having 



CP =y, .-. = cos-i ^ 



OE =y, .♦. = 



R* 



R' 



0'B=y, .-. = cot-i^, 



CE =y, .'.0= sec 



-il 



CB =y, o 'o = cosec 



.i2 



we have </0 = 
we have dip = 
we have d<p = 
we have ^0 = 
we have dif) = 



-4/ 



Rdy 
-Rdy_ 

^4/^ — i?a' 

— Rdy 
y^fZTR-^ 

dy 



PO =1', .*. = versin-i^, we have ai^ = - ; 

R ^2Ry—y^ 

y — dy 
0'0= V, .*. = coversin-i ^^ yyg have 0^0 = - , . 



EXAMPLES. 



1. dsin-^ x^ = 2xdx/ Vi — .r*. 

2. ^ sin-i 2;«: Vi — x^ = 2dx/ Vi — X*. 

3. d tan-i \ia + a:)/(i - a^)] = dx/{i + x^). 



DIFFERENTIA TION OF FUNCTIONS, 89 



4. d sin-i \sx + I)/ V2] = ^V ^i - 2;c - ^». 



5. a^ vers-i jt'^ = 2dxJ V2 — x'^ 



6. d Vsin~^ ;f = dx/2 Vshi~^ x{i — x^). 

7. d tan-i (— «/x) = adx/{a^ + x'*). 

8. dcos--^ [(i - ^2)/(i H--^')J = 2dx/(i-\-x% ^ 

9. ^ tan-i ( Vi - x/ Vi -\-x) = - dx/2 Vi — x\ 
10. ^sec-i [i/(2j:2 _!)] — _ ^.dxj Vi — x^. 



II. ^tan-i [x/ Vi - x''] = dx/ V 



:/ y 1 — X-, / 

I - x\ \/^ 



12. d cos~^ [x/{a — x)] = — adx/{a — x) Va^ — 2ax, 

13. d tan-i [x Vi7(2 + x)] = Vjdx/2{x' + -^ + l). 

14. i/cos-i Vi — jc-'^ = dx/ Vi — x^. ^ 



15. ^sec-l [n/ Va"" - x'^ = dx/ iV - x 



/^2 _ v-2 



[6. d cos-i [(4 — 3x'*)/x^] = — })dx/x Vx^ — i. 

I-], d — cot-i {x/a)/a = d tan-^ {x/a)/a = a';t:/(a'+ a:'). 

[8. a' tan-i [2^(1 + ^^)] = 2(1 - x'')dx/{i + 6;c' + x*). 



19. d sec~^ (x/a)/a = ^— cosec"! {x/a)/a = dx/x Vx^ — a'. 

20. ^ sin-i [(I - ;»:2)/(i + x^)] = - 2dx/{i + ;c'). 

21. ^tan-i [(i - x)/(i + x)] = dx/ii -f- jc"). 



22. ^sin-l [x/ Vi + ^i;''] = dx/{l 4- ;«:«). 



23. ^cos-i {2x — i) = — dx/ Vx{i — ;r). 

24. ^tan-i [2x/(i - ^*)] = 2dx/{i -{-x''). *4 

25. oT sin-i (1/ Vr^T?) = - dx/{i + ^»). 



26. fl^sin-i ^{i -x)/2 = — dx/2 |/i —x^. 

27. 0^130-1(1/ 4/x'^ — i) = —dx/x \^x^ — I. 

28. fl'cos-'(4^3 — 2)X) ■= — idx/ 4/1 — x2. 

29 ^ tan - 1 a^'^' = - «i'^ log a^. /x2 ( , + a^'^'). 



90 DIFFERENTIAL CALCULUS, 



30 



. d tan-i (|/i J^x'-x)=- dx/2{i -\-x''). 



31. d cos-'(l — 2x'^) — 2dx/\f\ — x"^ 



32. d /^" ^ = /'" -^ ./x/f/i - Jt^ 

33. d vers-^ (lA) = — dx/x \^2x — i. 

34. d\r vers-^0//r) — |/2ry — ^J — y dyj ^2ry —y^. 

35. t/;^^'^"'-^ = x^'""'-^(sin-' V-^-f log x/|/r=-"^Vx 



36. jj/ = 2 tan-' 'j/(i - ;t:)/i -\-x. (i — ;r)/(i + Jr) = tan'(>'/2). 



Hence, x^cosy, and ^ = — dx/\/i — x^ 

37. ;j/ = cos-i[(4 - 3x'^)/x'^]. Put (4 - 3x'')/x^ = z. 

Then dy/dz = (- i/|/i — z~)(dz/dx) = - 3/;>:V'x=' — I. 

38. Wheny = o, andjj/ = 2r, find the slope of the curve 



1. d log sin X = 

2. d log tan X = 



;t: = r vers— ^jv/'') — y2ry — y^. 

Ans. 00, and o. 

MISCELLANEOUS EXAMPLES. 
d sin X cosxdx dx 



sin X sin x tan ji: 

^ tan X dx 2dx 



tan X cos jr sin ;r sin 2x 



X dx 

3. d log tan — = 



sin X 



X dx = d\ — log . 

[_2 I + cos Xj 



(7t , X \ dx dx 

4" ~' ~ 



2/ sin (;r/2 -|-x) cos jc 
= sec X dx =^ d 



I , I + sin X 
log 



2 I — sin ;p 



]■ 



5. d — log cos X = d log sec x = tan ;r dx. 

6. rt'f?^ cos j; = ^^(cos X — sin x)dx. 

7. dxc^'"" -^ = /'^ ^ (I + -^ cos x)dx. 



DIFFERENTIA TION OF FUNCTIONS. QI 

8. dxe''''^ ^ = ^^°^ -^ (I - X sin x)dx. 

9. (at' sin (log x) = cos (log x)dx/x. 

ID. u = sin"* x/cos** X. .'. log u = m log sin ;i; — « log cos ;t:, 

, du ( cos jf , sin;c\ 

and — = \?7i \- n \dx. 

u \ s\n X cos xj 

[ms\n^-^x ns\n^ + ^x\ 
•*• ^^ = \ cosn-ix + cos^ + i;. j^' 

11. « = jf/^" •^. .-. log « = log ;i; ~f tan-i x. 

du = u{i/x + 1/(1 + x''))dx = /^""'^ (I + ^ + x')/{i -f- x^). 

12. tf'^^^" ^'"-^ = «x«-V« ^^° ^(i +;c cos x)dx. 

13. ^^^°^ -^ sin X = /°^ -^ (cos ;r — sin*^ x)dx. 

14. ^[sin nx/s\n^x'\ = — n sin(;? — i)x dx/sin^'^^x 

15. ^cos log (i/x) = sin log {i/x)dx/x. 

16. i/cos sin X = — cos ;t sin sin x dx. 

17. fl?'^^ log ;r = (?* log (ex^)dx/x. 

18. ^log (;»:/°^-^) = (i — X sin x)dx/x. 

19. fl'^'^ (log x)'^ = jc'^-' (log x)""-^ (m log jc + n)dx, 

20. ^x«^l°8^^ = (« + i);c«-^^^°^^^x. 



21. ^log \/(i -\- sin ^)/(i — sin x) = dx/cos x. 



22. ^log y (i — cos x)/{i -\- cos jr) = ^;i:/sin x. 

23. <^ sin tan x — cos tan x dx/cos^ x. 

24. ^ — log ^-^, — - -\ — tan-i ^- = — — — , 

25. d cos log sin ;r = — cot x sin log sin x dx. 

26. d sin~ f^sin x = t/x/2 4^1 -f- cosec x. 

27. a' log (x/a'') = log {e'^-^/a)dx. 



28. ^ log sin ;if = (a'jc'/sin ^ ^i — jc* 



92 DIFFERENTIAL CALCULUS, 

29. d sin~ (tan x) — sec* x dx/ \/i — tan'' x. 

30. d log cos" ^ = — dx/cos" X )/i—x^. 

31. ^ log tan~^;i: = dx/{i -^ x"^) tan""^;i:. 

32. d tan ~ log X = dx/x[i -\- (log xY]. 

33. 0^ cos [a sin~ (i/-'^)] = « sin (« cosec~ x)dx/x \/x^ — i. 

34. </ cot '^ (cosec x) = cos x dx/(\ -\- sin'* jf). 

35. d sin-'O^^"''-^) = /^"''•^^V(i+-=^')'^i-*''^''~'-*' 

Hyperbolic Functions, 
85. d sinh X = ^i(^^ - ^-^) 

= W"" + ^-'=)^^ = cosh X dx. 
d cosh X = d\(f + ^-*) 

= i(^^ — ^"^)^^ = sinh X dx. 
d tanh x = ^sinh j^/cosh x) — sech^ x dx. 
d coth X = ^(cosh jc/sinh x) = — cosech' x dx. 
d sech X = 4 1 /cosh x) = — sech x tanh x dx. 
d cosech x = ^(i/sinh x) = — cosech x coth x dx. 

EXAMPLES. 



1. ^4/cosh X = sinh X ^x/2y cosh x. 3. d log sinh x = coth x dx. 

2. d log cosh X = tanh x dx. 4. <^(;c — tanh x) = tanh" xdx. 

5. fl^[(sinh 2x)/4 + •^/2] = cosh'* j:^;f. 

6. a^[(sinh 2x)/4 — x/2] = sinh'* xdx. 



DIFFERENTIA TION OF FUNCTIONS, 93 

(x^ x^ \ 

X -\- . h 7— + ... I = cosh X dx. 
|3- li / 

I + I h -; h • • • ) = sinh xdx, 

ll ll / 



Inverse Hyperbolic Functions, 



86. d sinh-i x = d log (x + Vi + x') = dx/Vi + x' 
Let J' = sinh"^ x ; then 

X = sinh J, and (§ 85) dx/dy = cosh^. 
Hence (§ 73), 

^/sinh~^^ _ ^ _ ^ _ I 

^^ ~ cosh;; ~ Vi + sinh'^ ~ vT+^* 



d cosh-* x= d log (x + i/x^ - i) = dx/Vx' - i. 

Let^ = cosh"*^ ; then 

X = coshji^, and (§ 85) dx/dy = sinhj;. 

Hence (§ 73), 

^cosh-*jc_ I _ I _ I 

dx "sinh;;" Vcosh'y- i ~ Vx' - i* 

d tanh-i x = d - log i±^ = — ^(^ < i). 
2 *I— X I — x'^ ^ 

Let j; = tanh--':^;; then 

X = ta.nhy, and (§ 85) dx/dy = sech'^. 



94 DIFFERENTIAL CALCULUS. 

Hence (§ 73), 

dx.2,x\\C^ X III 



dx sech y i — tanh y i — ^ 

d coth-i X = d - log ^-i-^ = £^:r^(^ > ^)« 



Let J = coth~^ x\ then 

:v = coth J, and (§ 85) dx/dy = — cosech^j/. 
Hence (§ 73), 

d coth-^ X — I — I — I 



dx cosech^ y coth^ y — '^ x^ — \ 



. ,', ,, ii±i/i-x^ -dx 
d sech-i X == d log 



Let J = sech"^ x; then 

jc = sech jF, and (§ 85) dx/dy = -- sech j tanhj^. 
Hence (§ 73), 

d sech~^ X — I _ — I 

dx sech jv tanh J j^\/j _ ^2* 



I ± y I + x' — dx 
d cosech-i x = d log — 



X xVx^ 4- I 

Letjj^ = cosech"^ x; then 

X = cosech_y, and (§ 85) dx/dy = — cosechjv cothji^. 
Hence (§ 73), 



DIFFERENTIATION OF FUNCTIONS. 95 

d cosech"^ x — i — i 



dx 



cosechj/coth^ ^|/^^ -j- i' 



EXAMPLES. 



r. d cosh-1 {x/d) — dx/ Vx'i — a^. 



2. d sinh-i (x/a) =,dx/ Va" + x\ 

3. ^tanh-i (x/d) = a dx/ia"^ — x"^), (x < a.) 

4. ^coth-i (x/a) = a dx/{d — x-^), {x > a.) 

5. ^ tan-i (tanh x) = sech 2x dx. 



6. ^[^tanh-i^ 4-|tan-ix] = ^ logy _ '^ -f i tan-^ .r 

7. ^ [^ cot-i jf — ^ coth-i x] = ^ I cot-i^ — logy ^ J^ ^ 



8. ^[^x |/x^ - d" - \a^ cosh-i(x/a)] = i/x^ - d" dx. 



9. ^ [|x \/d + x^ + ia" sinh-i (x/a)] = ^d' + x* ^x. 

Geometric Functions. 

87. Differential of an Arc of a Plane Curve.— Let s 

represent the length of a varying portion of any plane curve 
in the plane XY. It will be a func- 
tion of one independent variable 
only (§ 18), which we may take to 
be X. 

Assume any point of the curve, 
as M, and increase the correspond- 
ing value oi X -^^ OF, by FF' = 
Ax. /\s = MM' vfiW be the cor- 
responding increment of s^ and A/ = QM\ 




9^ DIFFERENTIAL CALCULUS, 

rrr, /e e \ ^-f limit ^^ i- ch J/ilf' 
Then (§ 70, § 44) — = ^^"^'- — = hm 






ds 



= Vdx' + ^V^x. Similarly — = Vdx' + df/dy. 



Hence, ds = l^dx' + dy'.* 

The double sign is omitted because s may always be considered as 
an increasing function of x. 

That is, f/ie differential of an arc of a plane curve is equal 
to the square root of the sum of the squares of the differentials 
of the coordinates of its extreme point. 

If s were to change from its state corresponding to any 
point, as M^ with its rate at that state unchanged, the 
generatrix would move upon the tangent line at M\ hence, 
MT = \/dx^ -\- dy^ represents ds in direction and measure. 

In order to express ds in terms of a single variable and 
its differential, find expressions for dy in terms of x and dx^ 
or of dx in terms of y and dy^ from the equation of the 
curve, and substitute them in the formula. 

Thus, let s be an arc of the circle whose equation is 
x^ +y — 4. Solving with respect to j, and differentiating, 
we have 



^' = qi xdx/ r 4 — x^ . 



* The square of the differential of a variable represented by a single 
letter is generally written as indicated in the above formula, and is 
similar in form to the symbol for the differential of the square of the 
variable. Similarly, the «th power oidx is generally written dx^. 



Hence, 



DIFFERENTIATION OF FUNCTIONS, 97 

2 dx 



ds 



/ 



dx" ^ 



X 



'dx' 



X 



v.- 



X 



88. Differential of any Arc. Let s represent the length 
of a varying portion of any curve in space. It will be a 
function of one independent variable only (§ i8), which we 
may assume to be x. 




Through any assumed point of the curve, as M, draw 
the ordinate MN \ and through N^ the point where it 
pierces XF, draw NP parallel to Y. OP will be the value 
of X corresponding to M. Increase x = OP by PP' =^ l\x, 
and through P' pass a plane parallel to YZ, intersecting the 
given curve at M'. As = arc MM' will be the increment 
of s corresponding to the assumed increment of x. 

Draw the chord MM' and the ordinate M' K, Through 
M draw MQ' parallel to a right line drawn through N and 
K\ and through N draw NN' parallel to X. Then 
N' K — Ay and Q'M' = As will be the increments of 
y and z corresponding to A^ ; and we have 



chord MM'= V^Axf + {Ayf + (As)^ 



98 



DIFFERENTIA L CAL CUL US. 



Hence (§ 70, g 44), 
ds_^ limit 2.YCMM' ^ lii^it ch. MM' 



limit 


V{ 


AA;)' + (Aj.r + (A2r 


AJfS^^c 


) 


Aa: 


limit 





5 may be a curve of single or of double curvature. The 
increment A^ may or may not lie in the projecting plane 
of the chord MM'. If not, the projection of the chord 
MM' on the plane XY will change direction as Ajj: ap- 
proaches zero, but the above relations will not be affected 
thereby. 

Let «'', P' and y' represent the angles made by the 
chord MM' with X, Y and Z, respectively, then chord 
MM' / l\x ^= i/cos a'. Let a^ (3 and y, respectively, 
represent the corresponding angles made by the tangent at 
M. Then 

ds/dx = ^^J^o[^/^^^ ^'] ~ i/cos a, and dx = ds cos a. 

Similarly dy = ds cos /3, and dz = ds cos y, in which x, y, z 
or s may be considered as the independent variable. 

89. Differential of a Plane Area.~Let u represent the 
Y area of the plane surface in- 

M____j\/' eluded between any varying por- 

tion of any plane curve, as AM^ 
the ordinates of its extremities, 
and the axis X. 




DIFFERENTIATION OF FUNCTIONS. 



99 



Regarding u as a function of x (§ 21), let x — OP' be in- 
creased by P' P" — Ax. P' MM'' P" will he the correspond- 
ing increment of zi. Hence (§ 70, § 46), 

limit 



du/dx 



Ax-m^ o 



IP'MM"P"I Ax'\ =y, 



which gives du=: ydx. 

That is, the differential of a plane area is equal to the ordi- 
nate of the extreme point of the boufiding curve into the differ- 
ential of the abscissa. 

To illustrate, let u represent the 
area BAMP, and PR — dx; then 
du =ydx = rect. PQ, which ful- 
fils the requirements of the defi- 
nition of a differential (§ 68). 

Similarly, it may be shown that 
^^ is the differential of the plane area included between 
any arc, the abscissas of its extremities, and the axis of F. 

In case the coordinate axes are inclined to each other 
by an angle B, we have du = y sin 6dx, or du = x sin f^dy. 

In order to express du in terms of x and dx, "substitute 
forji^, or dy, its expression determined from the equation of 
the bounding curve. 

Thus, if ^y + b'^x^ = a^b'^ is the equation of the bounding 




curve, we have jF = — ^ c^ — x^ 
a 



and du = - Va^—x'dx. 
a 



90. Differential of a Surface of Revolution. — Let the 

axis of X coincide with the axis of revolution; and let 
BM = i- be any varying portion of the meridian curve 

* It is important to notice and remember that ydx is the differen- 
tial of a plane area bounded as described ; and that it is not, in 
general, the differential of a plane area otherwise bounded. 



LofC. 



100 



DIFFERENTIAL CALCULUS. 



in the plane XY. Through M draw the tangent MT, 

the ordinate MF^ and the right 
line MR' parallel to X. Let u 
represent the surface generated by 
s\ and regarding it as a function 
of X (§ 24), let X = OF be in- 
creased by FF' = Ax. MM' 
= As will be the corresponding 
increment of s ; and the surface 
generated by it will be the increment of the function u cor- 
responding to A^. Hence (§ 70, § 56), 




^^ _ limit ^^^' S^"- ^y ^'"^ -M^M' 



27Cy 



dx 



cos R'MT' 



Assume FR = dx\ then R' T — dy^ MT = dsy and cos 
R'MT=dx/ds. Substituting this expression for cos R'MT 
in above, we have 

u _ _Z_i. and du = 27ryds = aTTyf^dx" +dy«. 

Hence, the differential of a surface of revolution is equal 
to the product of the circum. of a circle perpendicular to the 
axis and the differential of the arc of the generating curve. 

Similarly, it maybe shown that 27tx^dx'^^dy'^ is the 
differential of a surface of revolution generated by revolv- 
ing a plane curve about the axis of Y. 

In order to express du in terms of a single variable and 
its differential, find expressions iox y and dy in terms of x 
and dXy or of dx in terms of jv and dy^ from the equation of 
the generating curve ; and substitute them in the formula. 

Thus, ify = 2px is the equation of the generating curvr, 

\^2px and dy = ^-=^. 
V2px 



we have y 



Hence, 



DIFFERENTIA TION OF FUNCTIONS. 



lOI 



du 



= 271 V2j>x\/dx' -^t^ = 2n{2px -{-p')^dx. 



2pX 



91. Differential of a Volume of Revolution Let the 

axis of X coincide with the axis of revolution ; and let BM 

be any varying portion of the 

meridian curve in the plane 

XY. Through M draw the 

ordinate MP^ and the right 

line MQ' parallel to X. Let v 

represent the volume generated 

by the plane surface included between the arc BAf, the 

ordinates of its extremities, and the axis of X. Regarding 

z/ as a function of x (§ 29), let x be increased by BB'= Ax. 

The volume generated by the plane surface PMM' P' will 

be the corresponding increment of the function v. Then 

(§70, §48) 



Y 


m' 


M 


^^^'^' 


P' X 


b/ 




P t^x 



dx 



limit vol. gen, by PMM'P' __ 

Ax ^' 



AJf«»-»0 



and 



dv = Try'di:, 



Hence, the differential of a volume of revolution is equal to 
the area of a circle perpendicular to the axis into the differen- 
tial of the abscissa of the 7?ieridian curve. 

Similarly, it may be shown that nx^dy is the differential 
of a volume of revolution generated by revolving a plane 
surface about the axis of Y. 

In order to express dv in terms of a single variable and 
its differential, determine an expression for jj^ in terms of x^ 
or of dx in terms of _y and dy^ from the equation of the me- 
ridian curve, and substitute them in the formula. 

Thus, if jc' + / — 2Rx = o is the equation of the me- 



102 DIFFERENTIAL CALCULUS, 

ridian curve, we have dv = 7t{2Rx — x^^dx ; or since dx 
= q= ydy/ ^R' -y\ dv = ^ nfdy/ VR' -/. 

92. Differential of an Arc of a Plane Curve in Terms 
of Polar Coordinates — Let r = /{?;) be the polar equation 




of any plane curve, as BMM\ referred to the fixed right 
line PD, and the pole P. Let BM = i- be any varying 
portion of the curve, and PM = r the radius vector corre- 
sponding to M. Regarding i" as a function of v (§ 19), let 
V be increased by MPM' — /\v. The arc MM' = As will 
be the corresponding increment of s. With Z' as a centre 
and PM as a radius, describe the arc MQ', Denote PM' 
by r'; then Q'M' = r' — r will be the increment of r corre- 
sponding to Az'. Through i^ draw the tangent^/', and 
the chords MM' and MQ' . Then (§ 70, § 55) we have 



dj_ 
dv 



limit a rc MM ' _ Xxxv^xl .//r^-^y .yA/dr" ^, 



Hence. ds = Vdr^+rMv^ 



DIFFERENTIATION OF FUNCTIONS. 



103 



Also, 

^'_ limit e^^'^iim-^'-^- 

dv Av^^o ^i, arc Q'M 



= r lim 



Q'M' 




ch. Q'M 

If the radius vector J^M coincides with the normal to 
the curve at M, the corresponding tangent to the arc 
MQ' will coincide with M T; 
and (§ 55) 

^ _ limit arc MM^ _ 

giving ds = r^^^. 

In this case dr = o, because 
the motion of the generatrix at P B\\ 

the point considered is perpendicular to the radius vector. 
An important example of this case is a circle with the 
T pole at its centre. 

v^ Let BM = ^ be any arc 

of a circle, and BCM = v 
the subtended angle. Then, 
since the radius is always 
normal to the arc, we have 
ds ~ rdv. 

That is, the differential of 
an arc of a circle regarded as a function of the corresponding 
angle at the centre y is equal to its radius into the differential 
of the angle. 

To illustrate, assume MCQ = dv ; then will the arc MQ 
— rdv. The direction of the motion of the generatrix at 
any point is along the corresponding tangent to s ; hence, 
by laying off from M upon the tangent at that point a dis- 
tance M T =^ ds =z rdVj we have ds represented in measure 
and direction. 




'104 DIFFERENTIAL CALCULUS. 

In order to represent graphically the general case when 
ds = Vdr'^ + r^dv^, let BM be the given curve, I* the pole, 
M the assumed point, and MPM' = dv. If r were con- 
stant, as we have seen in the case of a circle, MT' = rdv 




would be ds ; but, in general, ds is affected by a uniform 
change in r, in the direction FM^ equal to dr. To deter- 
mine it we have 

dv Az^^^Och. (2 J/ 

At T' draw T' T parallel to PM\ then T'T/T'M = 
tan T'MT = T'T/rdzK Hence, dr/dv — rT'T/rdv = 
T'T/dv, and dr = T' T. MT = ds=Vdr' -f- rW\there- 
fore, represents ds in measure and direction. 

In order to express ds in terms of a single variable and 
its differential, find expressions for r and dr in terms of i> and 
dv, or an expression for dv in terms of r and dr, from the 
polar equation of the curve ; and substitute in the formula. 

93. Differential of a Plane Area in Terms of Polar 
Coordinates. — Let u represent the area of a varying portion 
of the surface generated by the radius vector PM revolving 
about the pole P. Regarding zi as a function of v (§ 2;^), 



DIFFERENTIA TION OF FUNCTIONS. IO5 

let MFM' = Av. The area MPM\ represented by Hu, 
will be the corresponding increment of u. Hence (§ 47), 



du _ 


limit ^u_ 


r' 


and 


rMv 
du = , 


dv 


Az^m->o /\i) 


2' 




2 



To illustrate, with FM = r, describe the arc of a circle 
MQ = rdv corresponding to MPQ = dv ; then du = r''dv/2 
= area of the circular sector MPQ. 




du may be expressed in terms of v and d7\ by substituting 
for r its value in terms of v^ determined from the polar 
equation of the bounding curve. 

94. Motion. — When a point changes its position with re- 
spect to any origin it is said to be in motion with respect to 
that origin. 

In general, the distance from any origin to a point in 
motion continually changes, and is a continuous function 
of the time during which the point moves. 

When the distance changes so that any two increments 
01 it whatever are proportional to the corresponding inter- 
vals of time, the distance changes uniformly with the time. 

* Motion, without regard to cause, is generally discussed under the 
head of Kinematics, but many important applications of the Calculus 
involve motion, therefore some of the definitions and principles of 
Kinematics are here and elsewhere introduced. 



I06 DIFFERENTIAL CALCULUS. 

The point is then said to be moving uniformly^ or with uni- 
form motion with respect to the origin. 

If the distance does not change uniformly with the time 
the point is said to be moving with varied motion with 
respect to the origin. 

A train of cars moves from a station with varied motion 
until it attains its greatest speed, after which its motion 
along the track is uniform while it maintains that speed. 

With uniform motion equal distances are passed over in 
any equal portions of time, and with varied motion unequal 
distances are passed over in equal portions of time. 

Let s in both figures represent the variable distance from 
any origin, as A^ to a point moving on any line, as MNO \ 




and let t denote the number of units of time during which 
the point moves ; then s =f{t). 

li f(t) is of the first degree with respect to /, the distance 
s will change uniformly ; otherwise the point approaches 
or recedes from the origin with varied motion. §57. 

The rate of change of s^ regarded as a function of /, cor- 
responding to any position of the moving point, is called 
the rate of motion of the moving point with respect to the 
origin ; and since uniform motion causes i- to change uni- 
formly with /, the rate of motion, in such cases, is constant. 
§ 59- 



DIFFERENTIA TION OF FUNCTIONS. 



107 



In varied motion the rate varies .with /, and is therefo/e 
a function of /. 

95. If the differential of the variable is assumed equal to 
the unit of the variable, the differential of a function and 
the corresponding differential coefficient will have the same 
numerical value. 



Thus, if %- = 2, and dx 

' dx 



I inch, we 



have , dx^^ 2 
dx 



B 



inches. In such cases the differential of the function ex- 
presses the rate in terms of the unit of the variable ; and 
since it is more definite, it is frequently used instead of the 
differential coefficient. 

To illustrate, let s denote any variable distance regarded 
as a function of time, giving 
s—f{t). Assuming any con- 
venient length to represent 
the unit of t, we may, by sub- 
stituting s for y and t for x 
(§ 20), determine a line, as 
AM^ whose ordinate repre- 
sents the given function. 

ds 
If PR = dt represents one hour, —dt = QB represents 

the change that s would undergo in one hour, from the state 
represented by PA, were it to retain its rate at that state ; 
and is more definite than the corresponding abstract value 
of ds/dt. 

96. Velocity. — The differential coefficient of the variable 
distance from any origin to a point in motion, regarded as 
a function of the time of the motion, is called the velocity 
of the moving point with respect to that origin. 



y^ 




d8 

Q 

R T' 





P dt 



I08 DIFFERENTIAL CALCULUS. 

Representing the variable distance by s, and the velocity 
by v^ we have v = ds/dt. 

For the reasons given above, velocity is measured by the 
product of ds/di and the distance assumed to represent the 
unit of time. 

That is, the measure of the velocity of a point in motion 
at any instant, in any required direction, is the distance in 
that direction that the point would go in the next unit of time^ 
were it to retain its rate at thai instant. 

It should be noticed that the distance referred to above, 

and represented by s, may or 
may not be estimated along 
the line or path upon which the 
body moves. Thus, if a point 
moves from A towards B^ and 
the velocity at any point, as C, 
in the direction AB is required, 
the distance s is estimated along the path described ; but 
if the rate or velocity with which a point, moving from 
A to B, is approaching D is required, s must represent the 
variable distance from the moving point to D^ in order 
that ds/dt shall be the required velocity. 

Since velocity is a rate of motion, it is constant in uni- 
form motion, and a variable function of time in varied 
motion. 

97. Acceleration. — The differential coefficient of velocity 
regarded as a function of time is called acceleration. It is 
denoted by dv/dt., in which v represents velocity. Since 
acceleration is the velocity of a velocity, it is generally 
expressed in terms of the distance which represents the 
unit of time. 




i^^ 



DIFFERENTIA TION OF FUNCTIONS. IO9 

98. Angular Motion. — Let C be a fixed point, CA a fixed 
right line, and B a point in motion so that the angle ACB, 




denoted by 0, is changing. Then the line CB is said to 
have an angular motion with respect to, or about, C. 

Let s represent the length of the varying arc, of any con- 
venient circle, subtending 6^, giving B = s/r. 

Both B and s are functions of the time during which CB 
moves. 

Angular motion is uniform when any two increments of 
the angle, or arc subtending the angle, are proportional to 
the corresponding intervals of time ; otherwise it is varied. 

99. Angular Velocity.— The differential coefficient of 
any varying angle regarded as a function of the time is 
called angular velocity. 

Representing any varying angle by 6^, and its angular 
velocity by 00^ we have 00 — d6/dt. 

If s denotes the varying arc of a circle whose radius is r, 
which subtends ^, we have 

6 = s/r; hence, go = dB/dt = ds/rdf. 

That is, angular velocity is equal to the actual velocity 
of a point describing any convenient circle about the 
vertex of the angle as a centre, divided by its radius. 

It is customary in applied mathematics to consider the 
radius equal to the unit of distance used in any particular 
case. Angular velocity will then be measured by the 



no DIFFERENTIAL CALCULUS. 

actual velocity of a point at the unit's distance from the 
vertex. 

100. Angular Acceleration.— The differential coefficient 
of angular velocity regarded as a function of time is called 
angular acceleration. It is denoted by dod/dt^ when gd 
represents angular velocity. 

PROBLEMS. 

1. The side of a square increases uniformly 3 in. a 
minute ; find the rate per minute of its area when its side 
is 6 in. 

Let X = side of square in inches, and u = area = x^; then dx/dt = 
3 in. min. and du/dx = ix. Hence (§ 77), du/dt = (du/dx)(dx/dt) = 
3 X 2x = 6x sq. in. min. and {du/di)x = 6 = 36 sq. in. min. 

2. The radius of a circle increases uniformly .01 in. per 
second ; find the rate of its area when the radius is i in. 

Let r = radius, and u — area = Ttr"^; then dr/d:( = .01 in. sec. 
and du/dr = 27m Hence (§ 77), du/di = .01 X 27tr = .027tr sq. in. 
sec; and {du/dt)r = i = .027t sq. in. sec. 

3. Find the rate of the radius when the area of a circle 
increases uniformly at the rate of 2 7rr sq. in. sec. 

Ans. I in. sec. 

4. The radius of a sphere increases uniformly .0491 in. 
sec; find the rate of its volume when the radius is 1.5 ft. 

Ans. 200 cu. in. sec. 

5. The volume of a sphere increases uniformly 500 cu. 
in. sec; when its radius increases at the rate of 2 in. sec, 
find the radius. Ans. 4.45 in. 

6. The area of a rectangle increases uniformly 100 sq. in 
min. Its base and altitude are increasing at the rates of 
3 and 7 in. per min. respectively ; find the area when the 
altitude is double the base. Ans. 118.34 sq. in. 



DIFFERENTIATION OF FUNCTIONS. 



Ill 



7. The diameter of a circle increases uniformly 3 in. 
sec; find the difference between the rates of the areas of 
the circle and its circumscribed square when the square 
is I sq. ft. Ans. 15.45 sq. in. sec. 

8.* A man 6 feet in height walks away from a light 10 feet 
above the ground at the rate of 3 mi. per hour. At what 
rate is the end of his shadow moving, and at what rate does 
his shadow increase in length ? 

Let X = AM — distance from foot of light to man, y — AB = dis- 
tance from foot of light to end of shadow, and s = MB = length of 
shadow. Let ( = number of hours. 




Then we have c/x/dl—3 mi. hr. ; ^Z = io ft.; MC= 6 ft. 
and it is required to find dy/di and ds/dt. 

The similar triangles ABI^nA DCL give x ; y \: ^ 

y = 5x/2, and dy / dx == 5/2. Therefore (§77) 



DL=4 ft.; 
10; hence, 



dy/dt = (dy/dx){dx/dt) = (5/2)(3) = 7.5 mi. hr. 
Also, X : s :: 4 : 6 ; hence, j- = 3^/2, and ds/dx = 2/^> 
and ds/dl ~ {ds/dx){dx/dl) = 1.5 X 3 == 4-5 mi. hr. 

9.* A vessel sailing south at the rate of 8 mi. per hour is 
20 mi. north of a vessel sailing east at the rate of 10 mi. an 
hour. At what rate are they separating at the time ? At 
the end of i^ hrs. ? At the end of 2^ hrs. ? When are they 
neither separating from nor approaching each other ? 



* Rice and Johnson's Calculus. 



112 



DIFFERENTIAL CALCULUS. 




Let t = time in hours from the given epoch. 

Let AB = y = 20 — 8^ = distance of first ship 
from EC t hours after the given epoch. 

Let^C— x=io/=clistance of second ship from 
BA at the same time. 



Let M =AC 
Given, 



Vx^+/ = V400- 32CV+l64/a. 



dt 



hr. 



dx 
dt 



mi. 

10—-. 

hr. 



^ . , du 
Required, — 

dt 



— 160 -|- 164/ 



ldu\ mi. ldu\ _ 1 mi. ldu\ __ 

\^// = o^~ hr7' \di)t=^i'^K-]hr~.' \dtJt = ^o-^' 



1/400 — 320/ -f- 164/'* 

mi. ldu\ 

7 hr. \aiit = ^o 

The follovi^ing general outline of steps may assist the stu- 
dent in solving similar problems : 

1°. Draw a figure representing the magnitudes and direc- 
tions under consideration ; and denote the variable parts 
by the final letters of the alphabet. 

2°. Write, with tjie proper symbols, all known data ; and 
indicate the symbols for the required rates. 

3". From the relations between the magnitudes find an 
expression for the function whose rate is required, in terms 
of the variable. 

4°. Differentiate and determine values or expressions for 
the required rates. 

In case an explicit function of a variable cannot be 
found, make use of the principles in § 77. 

10. x^ — 2pz is the equation of q Xl 

a parabola OM. A point starting 
from O moves along the curve in 
such a manner that z = 16.1/^; in 
which z is expressed in feet, and / in 
seconds. Find the rate of x with 
respect to /. 




DIFFERENTIATION OF FUNCTIONS. II3 

■b dz 





dx 
dz " 


p 

^/opz 


dx 

dt 


dx 
~ dz 


^ df 



1/32. 2/ /2 di 



32.2t. 



Hence, ^ = — X -j,= --=== X 32. 2^ — 4/32.2/. 

yi2.2pf'- 

11. One ship was sailing south 6 mi. per hour, another 
east 8 mi. per hour. At 4 p.m. the second crossed the 
track of the first at a point where the first was 2 hrs. be- 
fore. How was the distance between the ships changing 
at 3 P.M. ? When was the distance between them not 
changing? Ans. 2.8 mi. hr. ; 3 hr. 16 min. 48 sec. 

12. A ship is sailing south 60° east, 8 mi. per hour ; find 
the rate of her latitude and longitude. 

Ans. 4 mi. hr.; 4I/3 mi. hr. 

13. A point P moves in a straight line away from a point 
B dX the rate of 8 mi. hr.; find its velocity with respect to 
a point C situated upon the perpendicular to the line BP 
through B and at 100 ft. from B when BP — 50 ft.; when 
BP — 150 ft. Ans. 8/V'5 mi. hr.; 24/V13 mi. hr. 

14. If the diameter of a sphere increases uniformly at the 
rate of i/io inches per second, what is its diameter when 
the volume is increasing at the rate of 5 cubic inches per 
second ? Ans. \o/y n in. 

15. If the diameter D of the base of a cone increases 
uniformly at the rate of i/io inch per second, at what rate 
is its volume increasing when D 
becomes 10 inches, the height being 
constantly one foot ? 

Ans. 271 cu, in. sec, 

16. The base of a right triangle 
is 4 mi. ; its altitude is variable and ^/_ 
denoted by j', and is the variable 
angle opposite to 7. Corresponding to _;^ = 2 mi. find the 




114 DIFFERENTIAL CALCULUS, 

rate of 0, first as a function of j, then as a function of tan 
0. Explain the difference between the two results. 

Ans. 1/5; 4/5. 
17. A train is running from A to B at the rate of 20 mi. 
an hour. The distance from ^4 to C on a perpendicular to 
AB is 2 mi. Find the rate of the angle at C included 
between CA and a right line from C to the train. 

Let (p = variable angle at C, and j = mi. from A to train. 
Then CA — 1 mi., and dyjdt = 20 mi. hr. 



y = CAta.n(p. .', (p= ta.n-'^ -p— and 



^/ yiC'+y 4+r* 



18. Find the rate of the surface and volume of a sphere 
when its radius decreases at the rate of 2 ft. per minute. 

Ans. — i67rr sq. ft. min.; — Zttt^ cu. ft. min. 

19. A ball of twine rolls along a floor in a right line at 
the rate of 4 mi. per hour. One end of it is 30 feet above 
the floor and is attached to the top of a pole. At what rate 
is the ball unwinding when it is 40 feet from the bottom of 
the pole on the floor ? Ans. 3.2 mi. hour. 

20. A ladder 20 ft. in length leans against a wall ; if the 
bottom is drawn out at the rate of 2 ft. per second, at what 
rate will the top descend when the bottom is 8 ft. from the 
wall? Ans. 10.5 in. sec. 

21. The side of an equilateral triangle increases at the 
rate of 2 in. per minute. Find the rate of its altitude, and 
the rate of its area when the side is 10 in. 

Ans. 1^3 in. mi. ; 10 1/3 sq. in. min. 



DIFFERENTIAl'lON OP PUNC7F0NS, I15 

22. Two straight' railways intersect at an angle of 60°. 
An engine approaches the intersection on one of the tracks 
at the rate of 25 mi. per hour, and on the other track an 
engine is leaving it at the rate of 30 mi. per hour. At what 
rate are the engines separating when each is 10 mi, from 
the intersection ? Ans. 2.5 mi. hr, 

23. A man walking on a horizontal plane approaches the 
foot of a pole 60 ft. in height, with a constant rate. When 
he is 40 feet from the foot how will the rate with which he 
approaches the top compare with that with which he ap- 
proaches the bottom ? How far will he be from the foot 
when he is approaching it twice as fast as he is the top ? 

^ Ans. 2 I/1/13, V1200 ft. 

FUNCTIONS OF TWO OR MORE VARIABLES. 

loi. The Partial Differential of a Function of Two or 
more Variables, with respect to one of the variables, is the 
change that the function would undergo from any state, 
were it to retain its rate at that state, with respect to that 
variable, while that variable changed by its differential. 

The Total Differential of a Function of Two Variables 
is the change that the function would undergo from any 
state, were it to retain its rate at that state, with respect to 
each variable, while both variables changed by their differ- 
entials. 

Any function of two variables which changes uniformly 
with each variable has a constant rate with respect to each, 
and its form must be some particular case of the general 
expression Ax -\- By -f- C (§ 64). 

Representing such a function by z^ we have 

z^Ax^By^C, (i) 



Il6 DIFFERENTIAL CALCULUS. 

Increasing x and y by their differentials, and denoting 
the corresponding new state of the function by z\ we have 

z' = A{x-\-dx)^B[y^dy)-^C. ... (2) 

Subtracting (i) from (2), member from member, we have 

z' — z= Adx + Bdy. 

Since the function z changes uniformly with respect to 
each variable, the total differential of it, denoted by dz, is 
equal to the corresponding change in the function. 

Therefore, dz = Adx -\~ Bdy. 

Adx is the corresponding partial differential of the func- 
tion z with respect to x ; and Bdy is the same with respect 
to y. 

Hence, the total differential of any function of two vari- 
ables^ which changes uniformly ivith respect to each^ is equal to 
the sum of the corresponding partial differentials. 

The total differential of any function of two variables 
which does not vary uniformly with each variable is not, in 
in general, equal to the corresponding change in the func- 
tion, but it is equal to the corresponding change of a func- 
tion having a constant rate with respect to each variable, 
equal to that of the given function at the state considered. 
In other words, the total differential is equal to that of a 
function which changes uniformly with each variable, and 
which has at the state considered its partial differentials 
equal to the corresponding partial differentials of the given 
function. 

Hence, the total differential of any function of two vari- 
ables is equal to the sum of the corresponding partial differen- 
tials. 



DIFFERENTIATIOM OF FUNCTIONS. HJ 

In a similar manner it may be shown that the total differ- 
ential of any function of any number of variables is equal to 
the sum of the corresponding partial differentials. 

In order to distinguish between a total and a partial dif- 
ferential the symbol 9 is used to indicate a partial differen- 
tial. Thus having z = f{x, _y), then 

dz = {'dz/dx)dx -\- {dz/dy)dy, 

in which dz represents the total, 'dz a partial, differential. 

EXAMPLES. 

1. d{xy) = xdy-\-ydx. 

2. d{2ax'^y — 2y^ + S'^-f^ — 5) = taxydx -\- c^bx'^dx -\- 2,ax^dy — ^ydy. 

3. d\{x ^y)l{x -y)-\ = [2{xdy - ydx)]/{x -y)\ 

4. dixYz") = 2f2^xdx + 2x'^zydy + HxYzdz. 

5. dt2in-\y/x) = (xdy--ydx)/{x'Jrf)' 

6. ^sin (xy)] = cos {xy){ydx-{- xdy). 

7. d log {xy) — ydx/x -\- log xdy. 

8 dy^^^^ =y^^^ [\og y co^ xdx-\- {sm xdy/ y)]. 



9, d vtrs\n~\x/y) = {ydx — xdy)/{y^2xy — x^), 
10. d sin {x -\-y) = cos (x -\-y){dx -\- dy). 



II. Deduce the formula ds =ydr^ + rVz/' (§ 92) from the formulas 



, [Anal. Geo.,] and ^^ = ^dx^ -\- dyU%Sy). 

12. One side of a rectangle increases at the rate of 3 in. per second 
and the other decreases at the rate of 2 in, per second. Find the rate 
of the area when the first side is 10 in. and the second 8 in. in length. 

Ans. 4 sq. in. sec. 

102. In order to represent ^^=(aV^jt:)^ji:H-(aV^)^-..(i) 
graphically, let any state of the function z be represented 
by the ordinate JVM of a surface. § 27. 



Ii8 



D IFFERENTIA L CA L Ct/L t/S. 



Through JVM pass the planes J/7V7? and MJVP parallel, 
respectively, to ZX and ZV. Let MC he the intersec- 

W 




tion of the surface by the plane MNR, and let MB"be the 
tangent to it at M. Let M£ be the intersection of the sur- 
face by the plane MNP, and let ML be its tangent at M. 
MH and ML determine the tangent plane to the surface at 
M. 

Assume dx = NR, and dy = JVF. Complete the par- 
allelogram NS and the parallelopipedon NQ. Produce 
RK, PA and SQ. 

Then, \.2.n KMH =dz/dx, and KH = {dz/dx)dx. 

tan AML = dz/dy, and AL — {dz/dy)dy. 

Draw HB parallel to KQ. Connect A and D by AD, 
and draw LT parallel to AD. 

Then, QD = KH, and DT=AL. 

Therefore, QT =^ {dz/dx)dx + idz/dy)dy = dz. 



DIFFERENTIATION OF FUNCTIONS. II9 

Zr is parallel to AD, which is parallel to MH. Z" there- 
fore lies in the tangent plane, and MT is the tangent to MB^ 
the line of intersection of the surface by the plane MNS. 

While X and y are changing as assumed, the foot of the 
corresponding ordinate passes with uniform motion from A 
to S, and QT vi the total differential required by definition. 

As in the case of a function of a single variable, the ex- 
pressions 'dz/dx and dz/dy are, respectively, independent of 
dx and dy, but for any state of a function of two variables 
there is no fixed total differential coefficieitt. In the figure. 



QT/QM= tan QMT = dz/ Vdx' + d/ 

is the total rate of change of z, corresponding to the values 
assumed for dx and dy, but any change in the relative value 
of dx and dy will change this total rate. For any values 
assumed for dx and dy, the total rate of change of the func- 
tion, and therefore its total differential, will be the same as 
that of t/ie ordinate of the line cut out of the tangent plane 
by the plane through NM, whose trace on XY makes with 
X an angle = tan"' dy/dx. 

It is now apparent that the function represented by the 
ordinate of the tajigent plane at M is the one with a constant 
rate with respect to each variable, whose total differential for 
the same values of dx and dy is the same as that of the 
given function at the state corresponding to M. 

dz/ Vdx^ -{- dy'' = tan QMT is the tangent of the angle 
which a tangent to MB makes with XV, and (§65) its 
numerical value is the slope of the surface at M along MB. 

From (i), we write 

dz/dx = dz/dx + (dz/dy) (dy/dx), 

(2) 

dz/dy = dz/dy + (dz/dx){dx/d_y), 



I20 DIFFERENTIAL CALCULUS. 

the first members of which depend upon the relative value 
of dx and dy. 

Draw 7"/^ parallel to SR. Then MW is the projection 
of MT upon the plane MNR, dz/dx = tan KMW, and 
dz/dy is the tangent of the angle which the projection of 
MT upon the plane MNP makes with Y. 

'dz/dx — tan KMH is the partial differential coefficient 
of the function with respect to x only, and is independent 
of dx and dy. It is equal to the tangent of the angle which 
the intersection of the tangent plane, and any plane parallel 
to ZX, makes with X or the plane XY. Whereas dz/dx = 
tan KMW is dependent upon the quotient dx/dy (Eq. 2), 
which has no definite value because x and y are independ- 
ent, and their differentials arbitrary. 

If, however, by means of a second equation (p{x,y) = o, 
x and y are related, thus making one of them dependent 
upon the other and determining dy/dx, then z becomes a 
function of a single independent variable ; its graph becomes 
the line of intersection of the surface z=^/(x,y) by the 
cylinder (p{Xfy) = o, dz/dx becomes the differential coeffi- 
cient of z regarded as a function of x and is equal to tan ^, 
d being the angle between the axis of X and the projection 
of the tangent to the graph on XZ. 

Similarly both members of Eq. i may be divided by dt^ 
t being any independent variable, giving 

dz/dt= {'dz/dx){dx/dt) + {dz/dy){dy/dt). 

The values of both members will now depend upon the 
values assumed (§ 67) for dx, dy, and dt. 

But if we also write (p{x, t) = o and ^{y, t) == o, thus 
relating x and y to /, and determining dx/di and dy/di, 
then z becomes a function of one independent variable, 
and the first member becomes the differential coefficient of 
z regarded as a function of /. 



SUCCESSIVE DIFFERENTIA riON, 121 



CHAPTER VI. 
SUCCESSIVE DIFFERENTIATIONo 

FUNCTIONS OF A SINGLE VARIABLE. 

103. In general the differential coefficient, and therefore 
the differential of any function of a variable, are functions 
of the variable and may be differentiated. 

Thus, having ax^ ^ 

dax^ = ^^ax^dx^ and dax^/dx = T^ax^. 

Differentiating again, denoting d{dax^) by d^ax^, read 
^^ seco?td differential of ax^," and representing {dxY by dx"^, 
we have 

d{dax^) = d^ax^ = Saxdx"^, and d'^ax^ /dx^ = dax. 

(iaxdx^ is the differential of -^ax^dx, which is the differ- 
ential of ax^ . daxdx^ is therefore called the yfri-/ differential 
of T^ax^ix and the second differential of ax^. 

Similarly, dax is the fii'st differential coefficient of Ty'^x'^ 
and the second differential coefficient of ax:\ 

Differentiating again and extending the notation, we 
have 

d(d''ax^) = d^ax^ —- Sadx"", and d^ax^/dx^ = 6a. 

6adx^ is theyfri/ differential of daxdx"^, the i"<fr^;z^ differ- 
ential of T,ax'^dx, and the //^/r^y differential o{ ax^. 6a is the 
jfirsf differential coefficient of 6ax, the second of T^ax^^ and 

the tJiird of ax\ 



122 DIFFERENTIAL CALCULUS. 

Representing the function x^ by j', we have j^ = x^. 
Differentiating, we obtain 

dy = nx^''^dx, and dy/dx = nx'^~^, 

for the ^r^/ differential and differential coefficient, respec- 
tively, of x^. 

Differentiating again, we have 

d'^y = n{n — i)x'^~^dx^^ and d'^y/dx^ = n(n — i)jc""^, 

for the second differential and differential coefficient, 
respectively, of x"^. 

Again, d^y = n(n — i){n — 2)x'^~^dx^y 

and d^y/dx^ = n(n — i)(n — 2)x'^~^, 

for the ^Ai'rd differential and differential coefficient 
respectively, of x'^. 

Similarly, the fourth, fifth, etc., differentials may be 
derived in succession. 

It should be observed that the symbol d^y/dx^, which 
represents the second differential coefficient, is the differ- 
ential coefficient of the symbol dy/dx, which represents the 
first differential coefficient. 



Thus, -^^ = dV-^\ I dx. 



Si.,i,„,„ g = .(g)//.. 

Successive differential coefficients are also called derived 
functions or derivatives, and are frequently represented in 
order by accents on the functional letter. 



SUCCESSIVE DIFFERENTIATION. 1 23 

Thus, if/(^) represents the primitive function, then 

f\x\ f\x\ r\x\ etc, 

denote respectively the first, second, third, etc., deriva- 
tives oi f(x). 

Other forms are also used. Thus, having jj^ =1 f(oc)y 



D^y, DxJ, Dxy, etc., 

represent the successive derived functions in order. 

Each successive differential coefficient or derivative in 
order is the rate of change of the immediately preceding 
one, and d'^~'^y / dx^~^ is an increasing or decreasing function 
of Xj according as d^y/dx^ is positive or negative. § 71. 

Let y =/(x) = x'^f n being entire and positive. 

Then dy/dx = f{x) = D^y = nx^'K 

d^/dx' =f\x) = Z>^> = n{n - i)x''-\ 
etc. etc. etc. 

dy/dx"" = /"(jc) = D^'y = n{n - i){n - 2) . . . 2.1. 

d''+'^y/dx''+' = r-^\x) = JD/+'y = o. 

It should be observed that the symbols dy/dx^, /"(•^)> 
Dx^, etc., serve only to represent expressions, and to 
indicate their relations to the primitive function. In the 
above d'^y/dx^ denotes that n{n — i)^""^ is the second differ- 
ential coefficient of x^. 

The differential of any order is the product of the cor- 
responding derivative and power of the differential of the 
variable. 

Thus, dy = {d''y/dx'^)dx^=f''{x)dx'' ... (a) 

represents a differential of the n^'^ order, and d'^y/dx'^=^f''{x) 



124 DIFFEKENTIAL CALCULUS. 

represents a differential coefficient or derivative of the 
n^^ order. 

Dividing both members of (a) by ^x"~\ dx^~^y etc., in 
succession, we have, in order, 

(ty/dx''-'^ = d^d^'-^y/dx''-^) = f''{x)dx. 

d^y/dx""-^ = d\d'-''y/dx''-^) = /''{x)dx\ 
etc. etc. etc. 

d^'y/dx''-'' = d\d''-''y/dx^-'') = f''{x)dx\ 
etc. etc. etc. 

d^y/dx = d^'-^idy/dx) = f'(x)dx''-\ 

From which we see that the product of a derivative of 
any order by the ^rsf power of the differential of the 
variable is the Jirsf differential of the immediately preced- 
ing derivative, and its product by the second power of the 
differential of the variable is the second differential of the 
second preceding derivative, etc. The product of a deriv- 
ative of the n^"^ order by the {n — i)"^^ power of the differ- 
ential of the variable is the {ii — \Y^ differential of the first 
derivative. 

EXAMPLES. 

\, y ■=. ax*, dy = ^ax^dx, d^y = \2ax'^dx'^y 

d^y =z 2i\ax dx^, d^y = 24a dx*. 

2. /(x) = (a- x)-K fix) = {a- x)-\ f'{x) r-. 2{a - x)-\ 

/^(x) = 2.3 . . . n(a — jr)-"-i, etc. 

3. J = sin X. Dxy = cos x, Dxv = — sin x, 

Dx^y = — cos X, L)x*y = sin x, 

etc. etc. 

The exponent of the power of a function is diminished by 
unity at each differentiation (§ 79), and when entire and 



SUCCESSIVE DIFFERENTIATION. 1 25 

positive it will finally be reduced to zero. Hence, algebraic 
functions which do not contain fractional or negative ex- 
ponents affecting the variable have a limited number of 
derivatives. All others, including transcendental functions, 
have an unlimited number. 

4. f{x) = ax^ + bx"^. 

f'{x) = 3^x2 -j- 2dx, f"{x) = tax -\- 2^, /'" {x) = 6a. 

5. y = ax^. 

dy ^ -h d-'y ^ -i d^y ^ -f ^ 

~^^\ax , _ = -iax , ^3 = 1-^ . etc. 

t. y ■= a^. 
Dxy = a^ log a, n/y = a^ (log a)^ etc., Dx'y = a^ (log a)». 



7. 


y- 


= cos X. 














dy 
dx 


= - 


- sin X 


■ = cos(x + 


ly 


d^y 
dx^ ~ 


— cos Jf= 


cos( X 


+ 


^)- 


d^y 
dx' 


= 5 


sin X - 


= cos(. + ^), 


etc., 


dy _ 

dx'^ 


: COSj X 


^ + 


v)- 


8. 


y-- 


— log • 


X. 
















dy 
dx 


I 
x' 


d'^y I 
dx'' ~ x^' 


d^y 
dx^ 


2 


d^y _ 
dx" ~ 


2.3 

x"^ ' 


etc. 








dy/dx^ 


= {- 


i)n-i\n 


- l/xn. 








9- 


<P 


— cos- 


-^u. 















d(p/du = - 1/ \/i-u\ d''<p/du^ = - u/{i - «2)3/2^ etc 

10. /{x) = sin mx. 

f\x) = m cos ;«jr, f"{^) = — ^"^ sin /«;r, etc. 

p\x) = (- i)'^w2« sin mx, /^^+\x) = (— i)'^w2»^+i cos mx. 

11. y = e^^. 

dv d^v d^v 

J_ _- ae<^x^ _JL — a'igax^ etc. . . . — ^ = a'^e'^^. 

dx dx'' dx'^ 

12. / = log {e^ + e-^). d^y/dx^ — - %{e^ - e-^)/{e^ \- e-^f. 

13. jF = cos mx. dy/dx''^ = m'"' cos {7?tx -\- nil J 2). 

14. y — cos'^ ;<:. dy/dx^ = 2^''^ cos (2;c -|- nit 1 2), 



126 DIFFERENTIAL CALCULUS. 

15. y = sin X. dy/c/x'^ = sin {x -\- mt/2). 

16. JJ/ = (l + x)/{\ —X). d^/dx^ =r 2| V(l - xY+^. 

17. jj/ = tan X. d^yfdx^ = 6 sec* x — 4 sec^ x, 

18. J = \/2px. d'^y/dx"^ = — //(2/>x)=*'2 — _^2/y. 

19. /(^) = x'/{i - x). /iv (^) = 24/(1 - xf. 

20. y = e^ cos ^. d^y/dx'^ = 2^1'^e^ cos (jc -|- nit/^. 

21. J = ± j/i?^ — ^2. ^ — — 



dx'' ± {R-' - x-'f^'' ± y 

22. f{x) = tan X -f- sec :r. /"{x) = cos x(i — sin x)-^. 

23. jf/ =;>r^. dy/dx^ = ^^(i + log xf + X* -1. 

24. /(x) = x^ log X. /i^ (x) =r 6x-i. 

25. ^ = sin-y d^x/dy* = {qy + 6/)/(i -y)7/2 . 

26. jK = log sin X. Dxy = 2 cos ar/sin^ ;»;. 



27. JJ/ = |/sec 2;f, d^/dx^ = 3(sec 2x)^'^ — (sec 2xyi^. 

28. j)/ = {x^-\-a'')ian-\x/a). d^y/dx^ — ^a^/{a? + x^- 

29. _j/ = sec X. d'^y/dx'^ = 2 sec^ x — sec x, 

d^y/dx^ = sec Jf tan jr(6 sees x — l). 



30. 


;^ = 


^Jl-lJog X. 


dy/dx» 


' = \n - \/x 


•• 


31. 


/(^) 


= (2ax)^'b. 


/"(2) = 


4(a-&)/^(^ _ 


^)a(a+&)/&/^2. 


32. 


>' = 


tan-i (i/x). 


.*. X = 


coiy. 






dy _ 
dx ' 


,. I 


- sin^y, 


I 






-i + x'^ - 


•■ (1+^ 


2)n/8 - Sin"j/. 




diy 
dx-" 


d sin^ y 

~ dx 


2 siny cosy dy 


= sin 2y sin^jv. 






dx 



d^y ^(sin 2j)/ sin^ j) sin 2jj/ 2 sin y cos ;j/ <^^ 4" ^i"^ y cos 2;j/ 2c/i/ 
^jf*^ dx dx 

= 2(sin jF)(sin 2y cos j -|- cos 2y sin }^{dy/dx) 
=r — 2 sin^_y sin 3^, 



12/ 



S UCCESSI VE D IFFEREN TIA TION. 

Similarly, d^yldx^ = |3 sin* j sin 4^, 

and d^yldx^ = (—1)'' \n — i sin'^/ sin ny. 

Since tan-^ x = 7t/2 - tan"' (i/x), 

we have ^"(tan-^ x)/dx'^ = (- i)«-'|W— £ sin»^ y sin ^zy, 

1 n/2 

or ^»^(tan-J ;*:)/^x»^ = (- jY-i \n - i sin (w tan-ii)/(i +;f2) . 



34. J = aV(«' + •^')' 

35. _y= aV2 + C"/+C". 

36. ;/=^ + ^(x-af/^ 
^j//^;p = 3^/5(x - «)2/5. 



^7/^;c = a + C", of V^;«^"^ = C- 
d'^y/dx'^ = - t>c/2SKx - dfl^. 



37. fx=x'^± X^/\ f'x = 2X±^X^'''l2, f'x = 2± 15x1/74. 

38. fx = e'/\ f'x = - //7x2, f'x = (2x + i)e'^yx'' 



39. fx =g 



■l/x 



/'x=e-'/yx'. f'x = e-''\Y - 2x)/x\ 



40. The relation between the time, denoted by /, and the distance, 
represented by s, through which a body, starting from rest, falls in 
a vacuum near the earth's surface, is expressed very nearly by the 
equation s =16.1^'^; s being in feet and i in seconds. Construct a table 
giving the entire distance fallen through in i second; in 2 seconds; 
in 3 seconds; and in 4 seconds; the distance passed over during each 
of the above seconds; the velocity and acceleration at the end of each. 



Time in 
Seconds. 


Entire Distance 
in Feet. 


Distance each 
Second. 


Velocity. 


Acceleration. 


I 
2 

3 
4 


16.I 

64.4 

144.9 

257-6 


16.I 

48.3 

80.5 

112. 7 


32.2 

64.4 

96.6 

128.8 


32.2 
32.2 
32 2 
32.2 



128 DIFFERENTIAL CALCULUS, 

41. Having j- = 5/^, find the velocity and acceleration when / = 
seconds; if = 3 seconds. Ans, F'i; = 2 = 3'/io/2. Vt=2>= sV^S/'^- 

At = 2 = 31/572/4. ^^=3 = 31/573/4. 



42. _)/=sin (w sin-i jr). dy/dx—m cos (w sin-^ '^)/Vi — ^'^' Hence, 

(i - x''){dyjdxf = m^ cos2 (m sin-i x) = m\i -/). 
Differentiating again and dividing by 2dy, we have 
(l - x^){d-^y/dx^) - x{dy/dx) + my = O. 

43. fx = sinh X. f"x = sinh ;f. 

44. fx = cosh X. /"x = cosh ;r. 

104. Leibnitz's Theorem — Let ji^ = uv, in which ?^ and v 

are any functions of x ; then (§ 75) 

dy/dx — 2^ dv/dx + Z' du/dx^ 
d'^y/dx'' = u d\^/dx' + 2{du/dx)(dv/dx) + z' d''u/dx\ 

d''y _ d^ du d'^v d'^u dv d^u 
dx' ~ ^dx' ^^Jxd?~^^~d?dx~^ ^dx'' 

in which the numerical coefficients follow the law of those 
of the binomial formula. By a method similar to that used 
in deducing that formula for positive entire exponents it 
may be shown that 

dx"" ~^ dx""^ ^dxdx''-^'^ 1.2 dx^ dx'' "» + ' ' * 

,n{n—\)...{n — r^\)d''ud''~''v d'^~'^u dv d^u 

^ |7 di" dx^-^"- "^ • * • ^^.r"-!' ^ "^ '''^' 

EXAMPLES. 

I. f ^ /^"^X 

u — <f«^, du/dx = ae^^, . . . d'*^u/dx^ — a^e^^, 
V = X, dv/dx = 1, ... d'^v/dx'*^ = o. 

d^y/dx"^ = ;m"-i^"^ + «"^«^;r. 



SUCCESSIVE DIFFERENT I ATION. 1 29 

1. y =L e^^x"^. d'^y/dx^ — a'^-'^ e"^\(r x'^ 4" "^nax -f- n{n — i)], 

3. jj/ = j;^ tan X. 

d^y/dx^ = 2x^ sec'^jc(3 tan^ x -\- 1) -\- i^x'^ sec^jc tan x 
-{- i%x sec^ X -\- b tan x. 

4. jj/ = ^«^z/. 

^^'^ \ ' ^x^ I . 2 dx^^ ^ dxnj 



FUNCTIONS OF TWO OR MORE VARIABLES. 

105. Successive Partial Differentiation. — A partial dif- 
ferential of a function of two or more variables is, in gen- 
eral, a function of each variable and may, therefore, be dif- 
ferentiated again with respect to each. 

Thus, if z =/{x,y) we write (§ loi) dz = 'T^^ ~l~ v~^' 

in which dz/dx and dz/dy are symbols for t/ie partial 
derivatives of the first order \\\\\\ reference to x and _;)^ respec- 
tively. 

Differentiating dz/dx with respect to x^ we write 



9(S/"''^ = <I®) = (1?) 



for the partial derivative of the second order taken twice 
with respect to x. 

Differentiating dz/dx with respect to j^, we have 

drdz\ ^ d'z 

dy\dxi dx dy 

for the partial derivative of the second order taken once 
with respect to x^ and then with respect to/. 



130 DIFFERENTIAL CALCULUS. 

Similarly, we write 

dx \dy I ~ dy dx* dy \dy ) ~ dy'^ ' 

for the other partial derivatives of the second order. 

Each partial derivative of the second order will, in gen- 
eral, admit of differentiation with respect to each variable^ 
and the differentiation may be continued, in general, to any 
required order. 

The notation adopted is as follows: 



dx\dxV~ dx'' 



dy\dx') ~ dx' dy' 



dx \dx dy) 

d fd'z \ 

dy\dx dy 1 



dy) dx dy dx 



dy\dx dy i dx dy^ 
etc. etc. 

8^ + ' 



dyXdx''-'' dy"-) ~ 



dy\dx''-''dy-'l ^jc^-^^^ + i* 

which represents the partial derivative of the {n -\- if^ order, 
taken {ii — r) times with respect to x and (r -\- i) with re- 
spect \o y. 

The numerator indicates the number of differentiations, 
and the denominator the order of the successive operations 
with respect to the variables. 

By multiplying the symbol for any partial derivative by 
its denominator we obtain the symbol for the corresponding 
partial differential. Thus, d'''^''z dx'''dy'/dx'"dy'' represents 



SUCCESSIVE DIFFERENTIATION, 13 1 

a partial differential of the {jn -\- lif^ order, taken m times 
with respect to x^ and n times with respect to y. 
Having 2^ ^ f{x,y, z), 

'du/dxy 'du/dy^ 'du/dz, 

represent, respectively, the partial derivatives of the first 
order, each being, in general, a function of x, y, and z. The 
differentiation may, therefore, be continued, and the suc- 
cessive operations indicated by extending the notation used 
above. Thus, 'd'^^'^^^'u/dx^^dy^dz^ represents the partial de- 
rivative of the {;m -\- n -\- rf^ order, taken ni times with 
respect to x^ n times with respect to j, and r times with 
respect to z. 

In a similar manner, a partial derivative of any order of a 
function of any number of variables may be obtained and 
represented. 

lo6. Partial differentials and their correspofiding deriva- 
tives are independent of the order of differentiatio7i. 

Assume z = f{x,y), and increase x by h, giving (§ 70) 



dz _ limit 
dx ~ ti-B-^o 



fix -i-h,y) -f{x,y) 
h 



] 



In this expression increase y by k, and we have (§ 70) 

dy \dx I dx dy 
,.^.^ r limit r /(x+ ^, y^k)-f{x. yJ^U)-\f{xJrh, y)-f{x, ;/) ]"jl 



_ J5"^''t Y f{x^h, y\-k)-f{x, yJ^k)-[f{x+h, y)-f{x, yyn 



(I) 



132 



DIFFERENTIAL CALCULUS. 



Similarly, increasing J^^ in the primitive function by k^ we 
write 

?£ ^ limit \ A^^y + k) - Ax,y) ^^ 
dy k^^o\_ k S 

from which, increasing x by h, we obtain 

§■)- '■■ - 



8_ 

dx 



lim 



It r limit r 



dy dx 
AxJrh, y-^k)-f{x-^h, y)-\_Ax, y\k)-f{x. 



f]] 



limit 



oL- 



/(^ J^h,yJ^ k)-f{x ^k,y)- [f[^, y^k) -f{x , y)\ 
hk 



]• 



which compared with (i) gives 



^Jt: ^ dy\dx 
From which we have 



/ dx\dyl 



_a^ 

dy dx' 



-d'z 


d 


dx^dy 


dy 




d r 




^;^:L 



_dx \dx )j dx\_dy\dx /J 

8 / a^ \"i ^ a'-g 



Similarly, we derive 

8^^ 



dx\ dy /J dydx^' 
d'z 



d'z 



dx^ dy dx^ dydx dy dx^* 
and, in general, 

d'^+'^z/dx'^dy'' = d'^+^'z/drdx'^. 
Similarly, having u = f{^x,y, z), it can be proved that 

'd'^u/dz dx dy = 'd^ti/dy dx dz, etc. 



SUCCESSIVE differentiation: 133 

Hence we infer that the order of diff.erentiation in all 
cases does not affect the result. 

EXAMPLES. 
I z — X sxn y-\-y s\x\ X . = (zos y -^ zos x =. 



dxdy dydx 

2. 2 = 2xiy^ -f x^y. = _ — = 1 2(^2 +;/2). 

dxdydx dydx'' 



3. z = x logy. 



d'^ I d'^ 



4. z = x^y -\- 4y^. = ^x 



dxdy y dydx 



dxdy dydx 



5.. = .an-MiV 3' 



Q\2 • 



y I dxdy dydx (^^ + y'^) 

t. z = sin {ax'^ + by"^). ■ ^ ^ = — abn^ixyY-'^ sin («x^ + ^v^). 

7. ^ =;'^. -"T-f = j/^-Hi + -^ log;/) = --^ 

8. 2/ = ^^2/^. — ^:^ = (i + 2,^yz + x-y^'')^^^'^ = 

9- ^ - ^2^37- -^^ - - 8-^^ (^2 _y^3- 

x^y d'^u 2x'^z 'd'^u 2x 

II. U = o -^ , — ■ ^ 

z 



). 2 = sin-1 — ). 



^ - ^^' dydz {a" - z^f dxdy a^ - z"' 

'd^u /\xyz '^^u 4XZ 



dxdz 



12. z = sin — . 


d'z 


2 , X , X X 

= -V Sin [- - cos — . 

y^ y y* y 


dydx' 


13. z = sin (x +_)/). 


d/ 


- sin (X -\-y). ^= -COS (x+y). 



134 



DIFFERENTIAL CALCULUS. 



14. z = cos (x — y). 



:5. z = \og{x -^y). 






sin {x A^ y). 



dx'> 



= — cos [x — y) 
dx^ dx'^ 



dy^ 



= — cos {x — y) 



dy^ 



= cos {x -\-y). 

= sin {x — y). 
- = — sin(x— _y). 



dx-" {x+yY^' dy^ {x-\-yf' 

107. Partial Differentials of a Surface. — Let A TL be 

any surface, and ABCD — it 2. portion of it included be- 




tween the coordinate planes XZ, FZ, and the planes DQR^ 
B f'S, parallel to them respectively. From § 26 we have 

Increase OP = ^ by FP' = h, giving (§ 70) 

'du ^ limit r /jx^h^y) -f{x,y) ~\ 
dx /^;^^o |_ /i J 



SUCCESSIVE DIFFERENriATION. 135 

Now increase OQ — y\>y QQ' — k, giving (§ 70) 



RJL^ limit ^^^oL J, J 

dx dy k-m-^Q I ■ 

limit 



] 

^ J'^'q V A^^h, y+k)- fix, y-\.k) - [/(^ + h, y) - f{x, j)] "| ^ 

In which 

f{x J^ h,y^k)= AEGI, f{x, y^k)= ABHI, 
f{x + /i,y) = AEFD, f{x, y) = ABCD. 

Hence, 
f{x ^h,y-{- k) -f{x, y-{-k):= A EG I - ABHI = EEGH. 
f{x + h,y) -Ax,y) = AEFD - ABCD = BEFC. 
f{x + h,y-\-k) -f{x,y-\- k) - [/{x + /i,y) -/{x^y)] 
= BEGH - BEFC = CFGH, 



Therefore (§ 50] 



?)\i _ limit CFGH I 



dx dy k-^-^Q ^^^ cos /? •' 

and a^udx dy/dx dy = wsec ft dx dy. 

io8» Partial Differentials of a Volume.— Let ATL 

(figure § 107) be any surface, and ABCD-ON = z^ a volume 
limited by it, the three coordinate planes, and the planes 
B>QJ? and BBS parallel, respectively, to XZ and VZ. From 
§ 30 we have v = /{x, y). 

By the method used in the last Article, considering the 
corresponding volumes instead of the surfaces, we obtain 



3\/ limit 

- — -- = /i-^^-^o 
dx dy j^^^^ 



fix Jrh, y+k) - fix, y-^k) - If^x-^h, y)~fix, y)^ 



]■ 



136 DIFFERENTIAL CALCULUS. 

In which 

f{x^h, y-\-k)-f{x,y-itk)-[f{x-{- h, y) -f{x, ;/)]=vol. CFGH-NM. 
Hence (§ 49), 

a^z; _ limit CFGH-NM ^^^ 
dxdy ^^o hk 

and 8'vdx dy/dx dy = zdx dy. 

109. Successive Total Differentiation. — Having z = 
/fej), §101 gives 

^^==1^^+!^^' (^> 

in which the total differential of the first order dz, and the 

corresponding partial differentials -^dx and -r dy, are, in 

general, functions of x andjj^. 

Hence, the total differential of the second order, denoted 
by d'z, is obtained by differentiating each term in the sec- 
ond member of (i) with respect to each variable, and 
taking the sum of the partial differentials of the second 
order. 

. . dz . 

Differentiating j-dx with respect to each variable, we 

have 

dy 



^&^)==ll^"^ + l-i^"^^- 



Similarly, d (|^^) = ^-dy dx + p^/. 



dy dx"'^ "''^ ^ dy' 
Therefore 



d^z = |!? dx^ + 2 v^ dx dy + l^^dy^ . . (2) 
dx^ ^ dx dy -^ ^ dy^ -^ ^ ^ 



SUCCESSIVE D IFFERENTIATION. 1 37 

Differentiating again, since 






dx' ' ^dx'dy ^ ' dy' dx ^ ' dy' -^ 

Similarly, formulas for the total differentials of the higher 
orders may be obtained, the numerical coefficients of which 
will be found to follow the same law as those of the bi- 
nomial formula. Thus the formula for the n^^ differen- 
tial is 

"^ -dx-"" ^""dx--^ dy^"" "^^^ 1.2 dx^-^df'^'' ^-^ 
otherwise written for abbreviation 

'"'=^'''+14'' • • • • ^"^ 

which form is not to be interpreted as usual, but as fol- 
lows : Expand as indicated, regarding each term as a single 

/ 9 \" 
quantity, and in the result replace each term, as ( ;i— ^-^ J » 

by {d"z/dx")dx'^^ and each combination, as 



138 DIFFERENTIAL CALCULUS, 

It is important to notice that in deriving eq. (2) from eq. 
(i) we write, in accordance with § 106, 

d'^zdx dy/dx dy = Z'^zdy dx/dy dx ; 

and it follows that having any expression in the form Pdx 
+ Qdyy in order that it may be a total differential, it is 
necessary and sufficient that 

dy dxdy dydx ~ dx ' 

This is known as Euler's Test. It determines whether or 
not the given expression is the result of the complete differ- 
entiation with respect to both variables of some other ex- 
pression. Thus, having 

2xdx -\-ydx -\- xdy -f- 2ydy, 
we obtain 

which shows that the given expression is a total differential 
of some function of x and_)^. 

Having 2x dx -\-ydx -\- 27 dy, the test fails and the expres- 
sion is not a total differential of any function. By differ- 
entiating x'^ -{- xy -\-y^, the student may confirm both 
results. 

The successive total differentials of any function of any 
number of variables may be determined in a similar man- 
ner. 



SUCCESSIVE DIFFERENTIATION, 1 39 

Thus, having u = /{x^y, z), then 

+ 2 1 — r^-^ 6^+2 - — ^dydz + 2-7 — r dx dz, 
dxdy -^ dydz-" dx dz 

etc. etc etc. 

9 , . 9 . , 9 .N'^ 



^"/^ = \—-dx 4- v-^v + -7-^^) 2^.' 
V^jc ' dy dz I 



Rule. Differentiate the function and obtain expressions for 
the several partial coefficie fits to the desired order. Substitute 
these in the proper formula for their respective symbols, 

EXAMPLES. 
I. 2 = x^y^. dH = txy'^dx'' + I2x^ydx dy + 2x'^dy'', 

3. 2 = ;tr'»_j/2. ^=2 = 2;'V;tr2 -j- %xy dx dy + 2;t:'^2^ 

d'^z =i I2ydx'^ dy -f- I2x^x^'. 
* Extension of the symbolic form {a), § 109. 



I40 DIFFERENTIAL CALCULUS, 

4. z = (x'-^+j/^l'S. 



^'^ = (;,^+y)3/^ (/^^' - ^^y^^^y + ^'^/). 



e^, z = <?(a«+&2/). d^z — ,f(«a;+6j/)[^V;r2 + 2ab dx dy -\- d^dy^]. 

6, 2 = x^y"^ -\- y^x^. 
dh = (6-r/ + 2y')dx'' -\- I2{x^y + xy'')dx dy + {bx^^y + 2x^)dv''. 



IMPLICIT FUNCTIONS, I4I 



CHAPTER VII. 

IMPLICIT FUNCTIONS AND DIFFERENTIAL EQUATIONS. 

IMPLICIT FUNCTIONS. 

1 10. With the exceptions considered in § 73 and § 77, 
differentiation hitherto has been limited to ^jc//zWV functions. 

Lety = ax, and assume x to be independent, then (§ 14) 
^ is an implicit function ofx. 

Solving with respect to _y, we have _>' = ± Vax^ which 
expresses y as an explicit function of x. 

Differentiating, we have 

dy^ -^ adx/iVax (l) 

Otherwise, we may write 

^(/) = ^^-^j •*• d{y^)/dx = a, . . , (2) 

in which (/) =/{y), and y = (p(x). Therefore (§77) 

djf) ^4/),±^ ±^ 
dx dy dx dx 

Substituting in first member of (2), we obtain 
2y dy = adx, . . (3) /. ^ = ± adx/2^/ax^ . (4) 

which result corresponds with (i). 

Examining (3) we see that it, and therefore (4), may be 
derived directly from y = ax, by differentiating (y) as an 



142 DIFFERENTIAL CALCULUS. 

explicit function of y, y, and in general dy^ being functions 
of X. 

That is, the equation may be solved with respect to y^ 
andy differentiated as an explicit function, or we may differ- 
entiate, regarding y as an implicit function, and then solve 
the resulting differential equation with respect to dy. 

This principle is general; for, having any equation con- 
taining two variables, x and jf, it may be written 

f[x,y) = (p{x,y); 

and regarding ji^ as an implicit function of x, each member 
may be regarded as an explicit function of x, equal to the 
other for all values of x; therefore (§ 72) their differentials 
are equal. It is not essential that the value of the depend- 
ent variable shall be expressed in terms of the other, but 
it is necessary to remember which is assumed as the de- 
pendent and which the independent variable. 

The advantage of differentiating without solving with 
respect to the implicit function beforehand increases, in 
general, with the degree of that function. 

EXAMPLES. 



1. ay^ — x^ -{- dx = o. dy/dx = ± (3^2 — 3)/2 ^a{x^ — bx^ 

2. «y + b'^x'' = a^^ dy/dx = - b^x/ay. 

3. (^ _|_ ay = 4bx. dy/dx = ± {b/x)^^\ 

4. cos y = a cos x. dy/dx = tan jr/tan y, 

5. cos (x -\-y) = o. dy/dx = — i. 

6. xy — jj/* = o. dy/dx = {y^ — xy log y)/{x^ — xy log x). 

7. (y- xy = xK dy/dx =2x ± c^x^''^/2. 

8. ^y — x'^-\- bx'^ = o, dy/dx = ± (3;^ — 2b)/ 2 V~a{x — b). 



IMPLICIT FUNCTIONS. 143 

„ , dv 3^^ — IX {b — c) — he 

ax 2\ax{x — b) {x -\- c) 

cfy x{x^ -\- 12X —lb) 



lo. x'^ — 2x^y — 2Jf' — 8/ = O. — = 



dx 2{x^ -\- 4) 



IT./- 2yVa' -{-x''^x^=0. dy/dx = x/ ^ a" + x"". 

12. y s\x\ X - X sin_j/ 4"! = o« 

14. ;«;/-_>/ + I = o, dy/dx = e^/(2 - y). 



dy sin y — y cos x 
dx sin X — X cos y 



a-[.^a'-y' x ^ ^ a' - y'' dy V . 

15. log J =0. - = ~ ^^-^,' 

By continuing the differentiation in a similar manner, expressions 
for d'^y/dx'^, and derivatives of the higher orders, may be obtained. 

16. x2/3+//3 = a2/3. dy/dx = -y/V^i/^ 

d^y/dx^ = (i/3x^/^)(i/y^/^+y/V^^/^). 



17. x =:r vers~i (y/r) - ^/2ry —y^. 

dy/dx = {2r/y - i)^/^ d-^y/dx' = - r/y\ 

18. ay-^V= -aH''. 

dy _ b'^x _ bx dy 



ab 



dx d'y a i/x"" - «^ ^^' «>' (x^ - a^f'^ 

III. Having ti = /{^^ y) = o, and regarding y as an im- 
plicit function of x, we write (§ 102)* 

* Also u may be differentiated w^ith respect to x by differentiating 
it as a function of x and _y (§ loi), and since y is a function of x 
diu/dy must be multiplied by dy/dx (§ 77). 



144 DIFFERENTIAL CALCULUS. 

du 'du 'du dy __ , . 

d^ ~ dx'^dy~dx~ °' • • * • ^^^ 

EXAMPLES. 

1. M = jj/3 — 3jj/j;2 _j_ 2x^ = O. 

dx -^ dy ^-^ 3y'-3x x-^y 

2. y"^ — 2xy 4- a'^ = O. dy/dx = y/{y — x). 

dy _ (2y -j- ^W 



3. y 4_ 3ay _ 4^2^^ _ ^2^2 _ o. 



^ji; 2y^ -\- 3a^y — 2a^x 



4. X - ax;/ -r ^^^^ ;/ - o. ^^ ^^^ _ ^^^.^ _^ ^^3 . 

dy 3x^ 4- Aax -\- a} 

5. y-i _ x^ - 2ax^ - a}x = 0. -f - ^-^—^ =?— -. 

•^ -^ dx 2y 

6. y^ — x\i — ;f*) = O. dy/dx = {x — 2x^)/y. 



dy/dx = 2x{x'^ — a})f3ay{y -\- a). 

8. {av - x'y - (x - 4)5(x - 3)6 = o. 

dy^3^. 5(x - 4y{x - 3)8 3(x - 4)°(x - 3)5 
dx 4 8(47 - x') 4(4:1/ - x^) 

Differentiating (i), remembering that 

dx\dx) dx^ dxdydx* 

d^ ldu\ _ 9V 9^^// ^ 
dx\dyl dydx 9y ^^' 



IMPLICIT FUNCTIONS. 145 

we have 

d? ~ dx'' "^ dx'dydx "^ \dydx ^ 9/ dxjdx dy dx"" ~ °' 
d'^u d'^u dy 'd'^ul dyV dudy _ 

Hence, 



dx' 



X^dx' "^ Vjc- ay ^jc "^ a/ UW J / aj^' ' * * ^^^ 

Substituting expression for dy/dx from (2), and simplify- 
ing, we have 

dx-" ~ l_dx' \dyl ^dx dy dx dy ^ a/ \dxl ]l \dy/ ' ^^^ 

(6) 



Xi 


dx 


> V4; g 


"^^ ^^ 




dx"! 


' ¥ 








EXAMPLES. 




I. 


x^ — 3«;r>/ 


+ ;.3^ 


0. 










dx 


= 3(x^ - 


- ay), 


-dy 


= 3(- 


■axJrf\ 




av 


= 6x. 


dxdy 


= - 


3«, 


3'« - 6^. 

9/ 








■ x^ d^y 
ax dx'' 


= - 


2«3 


xy 




{f- 


■axf 


If 


dy/dx = 


/(6)giv 


es 














dx 




2X 






~ 3( 


- ax -\-y 


^) " 


ax — 


f 



14^ DIFFEKENriAL CALCULUS 

2. x^ -{-y^ — 2dxy — a^ = o. 

dy/dx = {x - by)/{bx - y), 
d^y/dx"" = {b"" - i)ay{y - bxf. 



If dy/dx = o, dy/dx"" = — i/aVi — b\ 

3. ixy — y"" — a" =1 Q. 

dy/dx = y/{y — x), 
dy/dx"" = y{y — 2x)/{y — x)\ 

4. x^ + ^axy -\- y^ = o. 

dy/dx z= — {x^ -\- ay)/{y^ -f" ^•^)» 
dy/dx"" = 2a^xy/{y"" -|- ax)^. 

5. y^ — x^/{2a — x) = o. 

dy/dx = ± {^a — x) Vx/{2a — xy, 
d'"y/dx"" = ± 3aV(2« - xf Vx. 

Differentiating (3), we obtain 

~^ ^\_ dxdy "^ 3/ ^^J ^^"^ "^ dy dx' ~ ^' • • v7 J 
Similarly, it may be shown that 

d^~~d^ "^ • • • "^ ay ^^« ~ °' • ' ^^ 

in which the intermediate terms involve differential coeffi- 
cients of jF with respect to x of orders less than the n^^. 
112. Having any equation containing three variables 



DIFFERENTIAL EQUATIONS. 147 

x^ y and z^ one must be an implicit function of the other 
two (§ 9). Each member may, therefore, be regarded as a 
function of but two independent variables. 

The differential of each member is equal to the sum of 
its partial differentials; and since the partial differentials of 
the two members are respectively equal to each other, the 
total differentials are equal. 

It is not necessary to express the implicit function or 
dependent variable in terms of the others, but it is always 
important to distinguish it. 

In a similar manner it may be shown that the total dif- 
ferentials of the members of any equation are equal, re- 
membering that the number of mdependent variables is one 
less than the number of variables. 

DIFFERENTIAL EQUATIONS. 

II3« An equation which contains differential coefficients 
is called a differential equation. 

A differential equation obtained by one differentiation 
may, in general, be differentiated again, giving a differential 
equation of the j-^^:^;^^ order, and so on to differential equa- 
tions of the thirds etc., orders. 

Thus, having y — 2mxy -\- x^ ^^ a^, regarding x as the 
independent variable, differentiating, equating the results, 
and reducing, we have 

y dy — mx dy — my dx -\- x dx ^^ o, 
or dy/dx = (my — x)/{y — ?nx). 

Differentiating again,* we have 

* The importance of distinguishing between the independent and 
dependent variables becomes apparent at the second operation of 
differentiation — as in above dx is a constant, whereas dy = /{x). 



148 DIFFERENTIAL CALCULUS. 

dy^ -\-y (Py — mx dy — 2m dx dy -\- dx^ = o, 

or, dividing by dx*, 

(y — mx) dy/dx^ -\- dy^/dx"^ — 2 m dy/dx -}- i = o. 

The order of a differential equation is the same as that 
of the highest derivative it contains, and its degree is de- 
noted by the greatest exponent of the derivative of the high- 
est order in any terra; provided that all such exponents are 
entire and positive. Thus, 

{dy/dxY — a/x =^ o is of the ist order and 2d degree. 

dy/dx"^ -{-ay =0 is of the 2d order and ist degree. 

. {dy)/dx''y'' + M{d''-y/dx''-'Y~' + . . . = o is of the n^^ 
order and m^'^ degree. 

EXAMPLES. 

1. y 4- JT^ — ^2^ dy/dx = — x/y, d^/dx^ = — ryy. 

2. / = 2j?x. dy/dx =p/y, d'^y/dx'^ = - p''/y^. 

3 . a'y'' + Vx"" = aH"". dy/dx = - ^ V/" V- ^ W-^-^' = - ^*/aY. 

4. y - ^2„^ ^y^^ = « V31/2. d y/dx"" = - 2a*/gyK 

5.ax-y{x a). ________ __ _ ____^ _ . 

6, y = l>x\ : dy/dx = 'ibx''/2y. d''y/dx'' = 3Pxy^y^. 



7. / = 2;>x + r^x^. 



dy^ _ i±±J^ d'^y _ _f 
dx y ' dx'^ y^' 



8. xV=4«'(2«— j). dy/dx = — 2xy/{x'^ -\- 402), 

d '^y/dx'^ = 2i'f 3,'. ' - 4«')/(x^ + 4«')'. 



DIFFERENTIAL EQUATIONS. 1 49 

10. \og{xy)-{-x—y = a. {x — xy) £^ -\- y -\- xy = o. 

12. )/^ — 2;rj/-|-^^ = O. d'^y/dx'^ — y{y — 2x)/(y — x)K 

13. ^^ + 3<T^-^/ +jJ^^ = O. d'^y/dx'^ = 2a^ ^yKy^ + ^•^)^« 

,4. .'^ = 4«'(3.->'). £=-^-^- 

15. j/3 = 2aji;'^ - x^. d'^y/dx^ = - Sa^/gx^'\2a - xf\ 

16. x'^ — xy-\-J =0. d'^y/dx^ = 2 (i + lA^). 

17. \og(x+y)=x-y. d'y/dx' = 4{x-^y)/{x+y+iy. 

_ ^34. 5^2 _|_i2,r - 8 



18, x'^—2xy—2x'^-8y=o. 



dx^ ^ (^^+4)' 



It is important to notice the difference between the suc- 
cessive differentials of an independent expression which 
contains variables and those of the same expression limited 
to a constant value. Thus, suppose we nave (x"^ -\-y^) un- 
limited, and (x"^ +y) = «^ 

Then, 

d(x^-\-y^) —2x dx-\-2y dy, and d{x'^-\-f)=^2X dx-\-2y dy^o, 

d\x''+f) = 2dx'^2dy\2iXidd\x'-\-f) = 2dx''+2d/^2yd'y = o. 
d\x'-\-y')^o, and d\x''-^f) = A,dy d''y^2dy d'y^2y d'y=o. 
etc. etc. etc. 



I50 DIFFERENTIAL CALCULUS. 

In the first case both variables are independent, but in 
the second only one. The difference becomes apparent 
at the second differentiation. 

Equations derived by differentiating primitive equations 
or other differential equations are called immediate differ- 
ential equations. 

114. Differential equations also arise by combining suc- 
cessive immediate differential equations with each other 
and the primitive equation in such a manner as to elimi- 
nate certain constants^ or particular functions which enter 
the primitive equation. Thus, from 

y=^ ax -{• b we obtain dy/dx = a. 
Eliminating a^ dy/dx = [y — b)/xy 

which is independent of a. 

Differentiating again, we have d'^y/dx^ = o, which is in- 
dependent of both a and b. 

As another example, take the equation of a circle 

(x - aY -\-(y-by = J^' (i) 

Differentiating three times, we have 

2(x — a)dx + 2(y — b)dy = o. . . . (2) 

^^'^^^f^(y-l,)^y=o (3) 

2dydy ~\- dy dy -{- (y — b)d^y = o. . . (4) 
Dividing (3) by "(4), member by member, we have 

(dx^ + dy')/2idy d'^y = d^/dy, which gives 



DIFFERENTIAL EQUATIONS, I5I 

dx'^-df ld''y _ dv_d^ldy_ 

dx" I dx'~^dxdx'/ dx'' ^^^ 

dx'[\dx) "^ 'J ^\dxV dx ~ °' 

which is independent of a, b and R. 

And, in general, by differentiating an equation n times we 
obtain n differential equations between which and the prim- 
itive equation n constants or particular functions may be 
eliminated, giving a differential equation of the «*^ order. 

EXAMPLES. 

1. Eliminate e^ and sin x from jj/ = ^* sin x. 

dy/dx = e^ sin x -\- e^ cos x = jj/ -}- <?* cos x, 
d^y/dx^ = dy/dx -\- ^* cos x — e'' sin x; 
and since e^ cos x = dy/dx — y, 

d'^y/dx'^ — 2dy/dx — 2y. 

2. y = a smx -\- b sin 2x. 

dy/dx = a cos x -\-'2.b cos 2Jf, 
d^y/dx^ =. — a sin jr — 4^ sin 2x, 
d^y/dx^ = — a cos x — 83 cos 2x, 
d^y/dx^ = a sin x -|- i6(5 sin ix. 
Hence, ^V^-=^' + Sd^y/dx'' + 4^ = 0. 

3. j/3 — 2^;^;'^ = — a^. (^y'^{dy/dxy — 2^x^ dy/dx =. — ityx'*, 

4. TV — nx'^ — b = O. x^dy/dx'^ — x dy/dx = Sjj/. 
?.(!-]- ;r^)(T + r'^) = ax^ (t -1- x'')xr dy/dx = I -f-_j/^ 



152 DIFFERENTIAL CALCULUS, 

d. y z= a sin x — b cos x. d^y/dx^ ■= — y. 
y. y = ax -\- a — aK (-^ + i)dy/dx — (dy/dxY = ^• 

8. _y = (a -j- ^jr ).?<'»'. d'^y/dx'^ — icdy/dx = — c^y. 

9 jj/ = sin jf. {dy/dxY -\- y^ = i. 

ro. _)/ = (?^ cos jc. d'iy/dx^ = 2dy/dx — 2y. 

11. _;/ = sin log ^. ^t'^^ ^y/dx"^ -\- x dy/dx -f ^ = 0. 

12. y = log sin x, d^y/dx^ -\- {dy/dxy = — i. 

13. f = ipx + r-x\ yx^^ + "^t^)'" '^-^^ "^^^ " ""* 

14. y = a cos (^x -j- r). dy/dx"^ = — ^2^. 

15. y = sin~ijf. (l — x^)dy/dx^ — x dy/dx = o. 



lb y = ex '\- \/i-\-c^. y = xdy/dx + |/i + {dy/dx)*. 

17. ^ = (x 4" ^)^"*- ^/^^ — «j/ = e^^. 

18. ;j/=(r^-tan-ia;^tan-i ;p— I. (i -f- x'^)dy/dx -\- y = tan-*;ir. 

19. y = (ex -f- log X -j- l)~^. xdy/dx -\- y = y^ log x. 

20. ^'^ — 2CX — c^ = o. y{dy/dxY -\- 2xdy/dx = y, 

ly — c cos mx-\-c' sin mx. ,„ , , « , „ 

21. K / , ,x dy/dx^ + ;«V = O. 

{ y =: C COS (iWJC + <r ). 

22. >/ = ;r log {{c -\- c'x)/x\ x^d'^y/dx^ + (/ — xdy/dxf = O. 

23. jj/ = JT sin nx/2n -\- c cos nx -j- r' sin ;2;c. 

d'^y/dx'^ 4" '^l^ = c°s fix 

24. jj/'^ sin'^ ^ + 2ay + ^2 = 0. {dy/dxf + "^y cot xdy/dx = y*, 

25. :j/ = (^3; + e-^)/{e'^ — e-^). y"^ — \ -\- dy/dx = O. 

26. ^2j/_j_2ax^!/ -I- ^^^=0. {x" - i){dy/dxy = I. 



DIFFERENTIAL EQUATIONS. 153 

27. jf/ = ae^^ -{- be-'^^ . d'^y/dx'^ — c'^y. 

28. y— ae^^ -\r be-^^ -\- ce^ . d^y/dx"^ — "jdy/dx = — 6y. 

29. y={a-{-bx^x'^/2)e''-]-c. d^y/dx^ — 2d'^y/dx'^ -\- dy/dx =e^. 

30. y={a-\-bx-\-cx'')e^-\-d. dy/dx*-2,d^y/dx^-\-2d'^y/dx''-dy/dx=o. 



154 DIFFERENTIAL CALCULUS, 



CHAPTER VIIL 
CHANGE OF THE INDEPENDENT VARIABLE. 

II5* Having any expression or equation containing dif- 
ferentials, or derivatives, of y regarded as a function of x^ 
it is sometimes desirable to obtain a corresponding expres- 
sion or equation, in which y^ or some other variable upon 
which y and x depend, is the independent one. This oper- 
ation is called changing the independent variable. 

The principle deduced in § 73 enables us to make the 
change in cases involving the first derivative only. 

When differentials of a higher order are involved it must 
be remembered that we have written 



d_ 

dx 



(±]-^:i and ^(^]~^ 
\dx J~ dx'' dx \dxy ~ dx' 



in which x is the independent variable, and dx is a con- 
stant. 

Regarding both dy and dx as variable, we have 



f dy\ __ dx dy — dy d'^x , . 

\d^l- ^? ~* .... (I) 



and 

d_ 
dx 



d d ldy\ _ d (dxd'^y — dyd'x\ 
ix ' dx \dxl ~ dx\ dx' J 



__{dx d'y — dy d^x) dx-\--^[dy d'^x — dx dy) d'x , . 
- dx~' ' ^^^ 



CHANGE OF THE INDEPENDENT VARIABLE. 1 55 

which should be substituted for d^y/dx^ and d^y/dx", re- 
spectively, in order that the results may be general, that is, 
in which neither x nor jv is independent. 

If ttien y is made independent, we have dy ■=. constant, 
d'^y = o, dy = o, and (i) and (2) reduce to 

dy — dy d'^x d'^x I Idx V , 

^^ dx' ^ ~~df I \d^l ' • • • V3) 

d^y _ ^dy(d^xy — dydxd'^x 

r id''x\ d'xdx-\ /(dxV , . 

which may be used when the independent variable is 
changed from x toy. 



EXAMPLES. 
Change the independent variable from x to^ in the following: 



■' dx' + \dx) - ^' 



d'^x j.dx_ 
~df'^'d^~^' 



d^y , IdyV^ — d'^x , dx 

^'d^ + 'Ad-xf =°- -^;^ + ^-^^ = °- 

d'^y , dy^ , d'^x dx^ dx 

d'y , idyy dy d'^x , Idx^ 

5. (^/ + dx'^f/'^ -\-adx dy = 0. (I + dx^dy-^f^ - a d'^x/dy'' = O. 

If X or jl' is given as a function of a third variable, 0, which we 
wish to make the independent one, we first transform the given ex- 
pression, by means of (i) and (2), into its general form in which 
neither x hor y is independent, and then substitute for x, dx, d^x, 
d^x, or y, dy, d^y, dy, their values in terms of 6 and its differential. 



156 DIFFERENTIAL CALCULUS, 

6. Having t — x -\- x"^, show that 

— -(4^+1)^ + 2^^. 

Change the independent variable from jr to in the following 
equations : 

7. -^ -^ A — - — o, when x = cos 9. 

^ dx"^ I- x^dx^ I - x"^ 

5r + "> = ''• 



^'y , I ^ , , 2 






10. ^v'* ^- + ax-^ -X- dy = o, when ;i; = ^ . 

^4(._.)| + ., = o. 

11. Having jc = r cos B, a.ndy = ?- sin 0, change the independent 
variable from x to 9 in the equation 

^-v+d^l /d^ ; (^> 

From i) we have R = J ,, ^ -^^ ^ . ; 

and, since d'^6 = o, 

f/jr = cos 6 dr — r sin dd, 

dy = s\n B dr -\- r cos 6 ^0, 

d'^x = cos ^V — 2 sin B dr dB — r cos a^0% 

</V = sin rtTV 4- 2 cos //r r/0 — r sin a^0^ 



CHAl^GE OF THE INDEPENDENT VARIABLE. 157 

dx" -f dy"" = dr" + rV92, 
d^xdy — d'^y dx = rd'^rdQ - 2dr^dB — r^dB\ 
Substituting these values, we have 

12. Having x — a{<p — sin <p) and jj/ = <3(i — cos 0), change the 
independent variable in eq. {a) from jc- to (J). 

Ans. 7? = — 4a sin (0/2). 

13. Having jr = r cos 6, and jj/ = r sin 6, transform 

2; =: (;f^ — ydx)/{ydy -\- xdx) 
so that r will be independent. Ans. z = rdO/dr. 

14. Having dy/dx = 3(jr — 2)', show that d^x/dy* = — '2-/<^{x —2) ^. 

15. — -i — r -^ = o, when ;i; = sm 9 '; 

</ji;' I — x^ dx 

d^y _ 



i6- :7:r5 + :xn-;:2 x = O' ^^^^ ^r = a tan 



</;«:2 ' «« + ^'^ dx 



d^y 



PART 11. 

ANALYTIC APPLICATIONS. 



CHAPTER IX. 

LIMITS OF FUNCTIONS WHICH ASSUME INDETERMI- 
NATE FORMS. 

Ii6. The symbols 

O/O, 00 /OO, O.OO, 00—00, 0°, 00°, i" 

are indeterminate forms. 

When for a particular value of the variable a function of 
the variable assumes any one of the above forms its corre- 
sponding value is indeterminate. 

The limit of such a function, under the law that the 
variable approaches the particular value, is determinate. 

(§39-) 

In many cases this limit may be found by simple alge- 
braic methods, otherwise a method of the Calculus is gen- 
erally used.* 

117. Form 0/0. 

Let fx/<px be a fraction such that both terms vanish 

T • -J ^ J lii^it fx 

when X = a. It is required to nnd -— . 

X »-> a (px 

* Any operation by which this limit is determined is getK rally 
called the evaluation of the corresponding indeterminate form. 

158 



INDETEBMINATE FORMS. 159 

From eq. (i), § 62, we write 

, Ax^h)=fx-\-hf\x^eh\ 

^{x -{-h)=cpx^ hct)\x + d'h). 

Hence 

/{a + ^) ^ f{a + Oh) 
<p(a-\-h) (p\a^e'h)' 

from which, as h ^-» o, we have 

fa/cpa = /'a/(p'a, 
or 

limit f^ __ Y f'^ _ f^ 
xM-^a (px (p' X <p'a 

Similarly we may deduce 

limit ffi _ ,. f!^_ _ ,. f^ _ f^a 

^, limit A f""^ 

Z**^ and 0**^^ representing, respectively, the derivatives of 
the numerator and denominator of the same and lowest 
order, both of which do not vanish when x = a. 

If /"^ and (p^a are both finite, the limit is finite. 

\i f^a is zero, and <p"a is not zero, the limit is zero. 

li /""a is not zero, and 0"^ is, the limit is infinite. 

Ufa and cp^'a are both infinite, the limit is undeter-= 
mined. 

Therefore the required limit is the ratio, corresponding 
to the particular value of the variable, of the derivatives of 
the numerator and denominator, of the sa7ne and lowest 
order, both of which do not vanish or become infinite. 



l6o DIFFERENTIAL CALCULUS. 

Each ratio as obtained should be carefully inspected and 
factors common to both terms should be cancelled if pos- 
sible before proceeding to the next ratio. 

EXAMPLES. 



0, 'i(m>n. 

1, if ;;?=« 
00 , iim<.n. 



3. (sin x/x)^^^ = cos j;]^ = I.* 

4. (^--i)A]o = ^^]o=i. 

5. tan ^/^Jq = sec'* xH = i. 

6. {x^ - a^)/ix' - a^)\ = 3^V2^]« = 3«/2. 

7. (^2 ~ ^2)/(^ _ ^f^^ - 2x/2{a - X)\ = CO . 

8. {X - af'/ia - x)y\ = |(x - .)Vy|(;. - .)V4]^ = o. 

9. (i — sin x)/cos ^J^/2 = cos x/sin •^J^/g = o« 

10. (^^ - ^-^)/sin jc]^ = (^^ + ^-^)/cos jf]jj = 2. 

11. xVsin x]q = 2^/cos jt]^ = o. 

x^ I IX '~\ _ X cos ^ I _ cos ^ — ^ sin X I _i 
sin jr/ cos ;cJo 2 sin x Jo 2 cos x Jo~'2' 

X — sin x\. _ I — cos jf I _ sin x~^ __ cos ^"1 i 

I — cos x\ sin X I _ I 

sin'* ^ _Jq 2 sin ;r cos jr_j0 2* 

* Hereafter, for abbreviation, the forms f{x)^^^ and fx~\^ will fre- 

limit , ^ ^ 
quently be used to express {fx). 



INDETERMINATE FORMS, 



i6i 



15. 



<»*— 2 sin X- 



;(? Jo I — cos X Jo 

-1 = 



^^-|-2 sin x—e-=^ 1 
sin jr Jo 



e^ -|- 2 cos jr + 



^mx _ ^ma n ^ ^^ mx "j _ ( 00 when r > I. 

^ ' {x - aj \a ~ r(x — af^Ja " 1 o when r < 1. 

Limits of factors of the given or any derived ratio may be deter- 
mined separately (§ 36). 



\/x tan X ~\ k/^ "1 ^^" "^"l -^ 

The given or any derived ratio may be separated into parts (§ 35). 
18. j/^- i/a-}-Vx-a ~] ^ / i_ ^ I \ / -y "1 



2x\/x 2x\/x — a. 

~ 2xr x-a Ja ~ ^Ta 
tan£_— _sin^~| _ / sin x \ /sec x — i\~l _ sec ^ — il 

Jo w A""^^^ yjo~ ^^ J( 



sec X tan x 



x^ I _ sec^x-|-tan'*;r secjcj __i_ 
Jo 2 Jo~2' 

tan jf — sin or\ /tan ^\ /i — cos x\~l i — cos or\ 
"• —l?^ Jo = (— ) i--^-) J. = -7^ Jo 
__ sin x "1 cos X 

~(n- i)x''-'Jo ~(n- i)(n -2);r"-'Jo "" "^ ' 
21. log (I +x)/x']^ = 1/(1 + x)\ = r. 

In some cases it is advisable to transform the terms before apply- 
ing the above rule. Thus — 

sin X "I 2 sin (x/2) cos (r/2)~l . . -, 

22. = ~- — -—-- — = cot (x/2) L = 00. 

I — cos ^J 3 sin^ {x/2) ^ ' '_[0 



1 62 DIFFERENTIAL CALCULUS. 

ii8. Form oo/oo , 

If yiz = 00 = 0^, we may write 



~;r — ^r~r — ~-> and (§117) 

0« <i>al fa o ^^ " 



0« 0^/ /< 
limit r I / I "1 ,. r0'^ //'jt: "I 

^~.Ls/aJ="'°LwV7?J 

=-[(i)'(g)]- 

and since limits of equimultiples of two variables with 
equal limits are equal, we have, multiplying by 

<t)xf'xlfx (p'x,^ 

limit fx .. f'x 

xM-^a 0x cp X 

Similarly, we may deduce 

limit /^^_ f''x _ __ f'x _ f^a 

The method is therefore the same as in the preceding 
article; but by §71 we have /''^/0''^ =: 00/co, when 
fa/cpa = co/00 for a finite value of a. Hence for finite 
values of a this method will fail to determine the limit un- 
less factors common to both terms of some derived ratio 
become apparent or the limit oi /""x/cp^'x becomes otherwise 
known. 



Taylor's Calculus, §79. 



INDETERMINATE FORMS, 163 



EXAMPLES. 



log sin 2x \ _2. cot 2x _ /cos 2x\/sin x\/\ 
logsin;(;Jo cot x _Jo \cosx /\sin 2;i:/_Jo 

_ /2 cos^ JT — i\ / sin X \ sin ^cH 

\ cos X /\sin2x/Jo sin 2;ir_jo 

3 2J(rJo 



cos 
= 2- 



loor 



^ = - -4^ = ~ ^]o = O = ^ log ^"lo. 

\Jx Jo lAUo -^ -^ 

i/x l _ _i/f!_~| _ sin^r"! _ 
cot ^Jo~ i/sin* ;rJo"~ ^^ Jo~ 

log -^ "1 _ i/x "1 _ _ sin^ X [ 

cosecj;_Jo cot j; cosec x_| xcos^_Jo 

2 sin jr cos X I 

= -. =0. 

cos X — X sin ^_Jo 

logjc I sin^ jv 1 . -1 

5. ■ = =: — 2 sin ;«• cos JT L = O. 

cot^Jo X Jo -J» 

in which « is a positive integer. If n is fractional, the exponent of 
X ultimately becomes negative, giving the same result. 
7. Putting j^ = \/xy whence y »»-> o as x ^-» 00 , we have 

In some cases we may with advantage place x •= a ■\- h and sub- 
sequently make ^ = o. Thus ^x — a/ \/x^ — a^ reduces to 0/0, and 
the ratios of all derivatives of both terms become 00/00 when x = a; 
but putting jc = a -|- ^, we obtain 

\/~^^^^a "I _ /^V3 n _ /?i/i^ ~] _ 



GO. 



164 DIFFERENTIAL CALCULUS, 

%. 

119. Form 0.00 . 

If /^ = o and 0^ = 00 , we write 

which takes the form 0/0 or 00 /co when x = a^ and the 
limit may be determined by the method of §§117, 118 
with the same limitations. 



EXAMPLES. 

-'" ;r Jo — X Jo 

2. //^'^ = — = = CO . 

Jo x-^X 2 Jo 

3. .-VV],=;.V^V-],=0. 

4. e-^ log x]^ = log x/e''\ = (i A)A^^ = O. 

5. sin jir log cot x\ = (sin jr/x) ;v log cot x\ = log cot jf/(i/;i;)T 

= (x'^/sin^^)(i/cot x)^^ — o. 

= I — \j/{x^ + i)]^ = I, when tan-i(i/oo) = O. 

7. sec J? (:v sin x— 7t/2)'J^^^ = (;r sin x — 7t /2)/cos ■^J^/2 

= (a- cos jc -{- sin .jf)/ — sin x \, = — i. 

8, log (2 - i), tan ^]^ = log(a - ^) /cot ^]^ 

— — I / — 7t/2a ~| _ £ 

~~ a(2 — x/a)/ sin^(7rx/2«) J^ ~ it' 



INDETERMINATE FORMS. 1 65 



10. x'^{\ogxf\^ 



— «/x(log xf'^^_\ 



.], 



tnx 
i/(log xfA( 

which remains indeterminate under the method; but placing x — e~^, 
whence x^{\og xf = (— ify'/e"-^ and ^/m-^co as x-m-^o, we have 
(example 7, § iiS) o for the limit. 

120. Form 00 — 00 . 

\i fa — <pa= ^ — 00 , we write 

which becomes 0/0 when x = a, and the rule of § 117 will 
apply after the given expression is placed under the form 
indicated in the second member. 

EXAMPLES. 



I. cot X 

X 



Jo \cot X II cot jrjo ^ ^'^"^ ^ Jo 

smx — xcosx _ X sin X _ sin jr-|-jtr cos ;tr _ 

;t: sin ;r Jo ~ sin X 4--^ cos X Jo 2cos jr— ;r sin jrJo~ 

In some cases the desired form may be obtained by a more simple 
transformation. 

i/~2 "I I — 4/1 — a/or\ ^ "] a 

2. X —yx^ — ax I = *- L— = — ■ -. = -. 

-^°° lA Joe 2fl-aAj^ 2 

3. tan ^ - sec x\/^= (sin x-i)/cos x']^^^ = -cos ^sin x']^^^ =0. 

2 _ I ~| _ 2{x - I) - jx'' - i) "1 _ 2— .y-i n 
c'- l~;«r-iji~' (x-^-i)(^-l) Ji~ ^=^-1 Ji 



4 . 



— I 

2X 



5. X tan jf — ;r sec x/2^^, — {x sin x — ;r/2)/cos -^J^/j 

= (x cos x-\- sin ;f)/— sin x^^j^ = — i. 



1 66 DIFFERENTIAL CALCULUS, 

121. Forms o*, oo*, i*. 

If {fa)'^''.— o*^ or 00^ or i°°, we write 

log (/x)*^ = (px log/x, whence (/xy^ = ^<^^io&/^^ 
and 

^^^^^ (/x)*^ = lim e'^^ ^°s -^-^ = (?'^'" ['^■^ loff^-*^] 

in which (pa log fa = o.oo in each of the above cases, and 
the method of § iig will apply after the given expression 
is placed in the form- indicated. 

EXAMPLES. 

3. x'n =,10^-/-] =.iAi =1. 

tan'^n tan '^ log 6-^)"] ^ 

5. (2 - x/a)^'^ Ja= ^ ^"^ J a ~ '^' 

6. cot ^sin ^J^ — ^sin ;« log cot ^ j^ _ j. 



^U^^ log a + 3»" lo g ^ + . . 
^ a«^ + <5«^ + . . . 



J^^^log(a^...«)^^3, . .«. 



122. Evaluation of Derivatives of Implicit Functions. 

Derivatives which for particular values of the variables as- 
sume indeterminate forms may be evaluated as in the pre- 
ceding cases. 



INDETERMINATE FORMS, 1 67 

EXAMPLES. 

1. y^ -\- x^ = ax^. 

Whence ^1 ^ ?^^J=Jf!l = ^Hi^l . 

^•^Jo.o 3/ Jo.o ^y{dy/dx)_\^,^ 

Hence, {dy/dxf^ . ~ °° ' ^"^ ^/^-^Jo . = ^ °° •"'"' 

2. x^ -\- y^ = 3axy. t/j//^x J^^ ^^ = o or 00 . 

3. r^ + 3«/ — 20^:7 = ax"^. dy/dx~^^^ = i or — 1/3. 

4. (^2 +y2)2 ^ ,,.(^2 _ y). ^^./^^J^ . = ± I. 

5. w = ;V^ 4" 3«V^ — 4^*^x7 — a'^x^ = O. 

'du/dx = — ^(i^y — id^x. 'du/dy — 4jj/' + 6a V "~ ^x. 



ay 
dx 



~j _ ld^y-\-d^x n _ 2a-{dyldx)-\- d^ ~j 

Jo.o ~ "^y' + 3«V - 2a'^Jo.o~ (^y-* + 3«')(^j/^-^) - 2^2 Jo , 

n . Hence (#)-i^1 =i 
2 Jo.o \dx I 3 ^^Jo.o 3 



_ lidvldx) -|- 
~" 2,{dy/dx) — 

and ^v/^^]o.o=(2 ± 4/7) /3. 

t. u- ay - a'^x^ -x'^=o. dyldx\ <, = ± I. 

7. « = X* + rtxV — ay^ — o. 4j//^xJq = o or ± I. 

8. « = oyS _ ^^2^ _[_ ^4 _ o. ^^'/^Jfjo . = ° °'' ± I^V^. 
g. M = jtr* — a2_;^y -f-y = 0« ^/'^'^Jo ~ ° ^'^ ^■'• 

10. M = JT* 4- >/3 -f «3 _ ^^j^y _ o. ^y/a'x]^ ^ = 1- (I ± 4/^). 

11. « = ^/s — 3axjj/ + jc* = o. </j//fltxJo .^ =r — o or 00 . 

12. « = ^* + '2.ax'^y — ay3 = o. dy/dx L ^ = o, or ± |/2. 

* ^r/c/x 1^ ^ has two limiting values for the same value of the vari- 
able, hence it is discontinuous at the corresponding state. 



l68 DIFFERENTIAL CALCULUS. 



MISCELLANEOUS EXAMPLES. 



3 

X 



log x~\ e^ — <?-^ — 2x "1 

-^— =0. 2. r =2. 

X _\^ x-%\nx Jo 

log f 1 ^ J (^-i)tan'^. y"] ^ ^ 

iji ' -^^ Jo 



I — cos X 



1 I 1 

= - , 6. sec jc — tan x \ = o. 

Jo 2 JV2 

^ -■ "1 ^ - COS- Hi — ^n 

7- -*8^ = o. 8. ■ — - = I. 

J 00 ^2X-X^ Jo 






a^ — b'^ 



]10gU_-f-_«£C)-l 

= «. 10. tf ^ = (f«. 

00 -^0 

« ^1 /tan ;t:\ V^n » 

;^-iog.-l ^ ^ (L^zilI = _ i. 

^lA Jtt ^ tan^^ Jo 2 

15. = O. 16. = CO. 

X Jo --^Jo 

19. (I + ax)^^ = ^. 20. log (^) /.]^ = 1. 

21. (I + ..)^J^= I. 22. ^^^^J, = log (7). 

log^l 24 ^A^ n -^ 

23. -^ = o. 24. ^^ - 8 • 

•^ Joo cot 

2 -"o 



= log -7 26. 2^ sin — = a, 

Jo ^ -^ 2^J^ 



/ , cosec2 fA:~j " l?a COt X + COSec JT — iH 

27. (cos ax) \ — e *•. 28. ; — = \ 

Jo cot^-cosecx+ij^/^ 



29. ^^"^^1 = I. 

gx ^sin x\ 

31. : = I. 

X — sin jrjo 

= 3«. 

\ ax — a _la. 



INDETERMINATE EORMS. 

30 

32 



169 



I j:); 

log X log x_ 



33 



34. 



35- ^ 

jf3 _ 2;t- 



^ ^"1 ^ L. 

' X — \ logx Ji 2 

;f3 — 3x +2 ~| _ 

x" - tx^ + 8;»: - 3J1 ~~ "'* 

^3—1 ~1 , tan «x — ft tan jf~l 

— ; = 3. 36. -. : = 2. 

2;tr^ -[- 2x — I w sin ^ — sin w^ j^^ 

37. tan Vlog (^ - V2)];r/2 = - «^ • 

38. (jc — rt 4- V2^x - 2a^)/\/x^ — af^^ — I. 

39. x/cO\. X — 7t/2 COS •^l;r/2 = — !• 

40. (i — x) tan (;rx/2)l^ = 2/7t. 

41. (aA-2 - 2acx + ac')l{bx'' - 2bcx + ^^2)^^ _ ^/^^ 

42. (i — sin j; + cos x)/(sin .a; -|- cos x — 1)1^/2 = !• 

43. (a - jr — a log a 4"^ log x)l {fl — ^2ax — ^^) J^ = — 1, 

44. {x sin x - n/2)/cos -^J^/j = — i. 

45. {e^ - e-^ - 2) sec x/x'^'\^ = 4. 

46. \ ^2 — sxti X — COS ;c)/log sin x\^,^ = — 1/2 |/2. 

47. [(^ - ^-=^)' - ^xXc^ + ^-^)]A''], = - 2/3. 

48. [(jf — aY/{e'' — ^«)]^ = o or ^-« or 00 , according as « > I, 

n = 1, n < 1. 

49. 1/(1 + ^lA) 4- ^VV^(i + ^ia/]^) = o. 

50. 1/(1 + e-y^) — e-y^x{i + ^-iA)']o = I. 

X + cos jr I I + cos x/x 

51. : = — ■ — r 7- = I. 

X - sin xj^ I - sin x/^ J^ 



170 DIFFERENTIAL CALCULUS, 



CHAPTER X. 
DEVELOPMENTS. 

123. The development of a function is the operation of 
determining an equivalent finite or convergent-infinite series. 
When this can be done, the function will be the sum of the 
series, which, in the case of a convergent-infinite series, is, 
also, the limit of the sum of n terms as n increases without 
limit. 

A convergent series having a given function as a limit 
may be used to determine approximate values of the func- 
tion, the degree of approximation depending upon the ra- 
pidity of convergence and the number of terms considered. 
A divergent series should not be employed in finding ap- 
proximate values of a function, or in the deduction of a 
general principle or formula. 

Let S represent a function giving 

.S" = 2/, -f 2^2 + ^3 + • • • + ^« + etc. 
Denote the sum of the first n terms by S^ , and the sum 
of the following terms by R^ called the remainder; then 

The series is convergent if R is an infinitesimal as n in- 
creases without limit, in which case S is the limit of S^. 

When S is the limit of S^ as n increases, it is also the 
limit of Sn - 1 , and we have 
limit 



Sn — Sn-x = lim Un — C 



DEVEL OP MEN TS. 1 7 1 

That is, in a convergent series the «*** term is an infini- 
tesimal as n increases, but the converse is not necessa- 
rily true unless Sn has a finite limit under the law; for 
lim Un =^ o =^ lim [6'„ — '^"«-i] may occur when 6" = oo. 
Therefore a series is not necessarily convergent when 
the n}^ term is an infinitesimal as n increases. 

124. Taylor *s Formula has for its object the develop7?ient 
of a function of the sum of two variables into a series arranged 
according to the ascending powers of one of the variables with 
coefficients which are functions of the other. 

Assuming an expansion of the proposed form, we write 

f(x^r h) =^ X,^ XJi^ XJi' -\- . . . J^X^^^h^^R, {b) 

in which X,, X^^ etc., are functions of x to be determined, 
and R the remainder after n-\- \ terms. 

^ = o gives fx = ATj. 

Placing x-\- h^ s^ and differentiating, first with respect 
to X and then with respect to h^ we have 

df{x + h)ldx = Ws/di) (ds/dx\ (§ 77). 

df(x + h)/dh = (dfs/ds)(ds/dh). 

But ds/dx = ds/dhy hence 

df{x + h)/dx = df{x + h)/dk 



172 



DIFFER EN TIA L CAL CUL US. 



Hence, differentiating the second member of {b) first 
with respect to x and then with respect to h, we have 



ax ax ax ax 



^^^• + ^^^" + ^^- 



= X, + 2X,h + ^X.h'^ + ^X,h' + etc. + nXn+,h-^-V etc., 

which is an identical equation, and by the principle of in- 
determinate coefficients we have 



dX, _ 



dX. 



V ^^3 XT <dXn „ 

2^3 > ~r: — 3^4 > • • • ~r~ =nXn+u etc. 



dx ^''' dx """ ^jc ^"4.--- ^^ 
Since X, =/r, dXjdx — fx^ .'. X^ — fx. 



Therefore 



-'=/"^ = 2X3, and X,^\f"x, 



dX. 

dx 

dX, 

dx 

etc. 

dX. I 



= i/'"^ = 3^o and ^^ = -^/":^. 

2-3 

etc. 
f" x—nXn^u and ^«+i =p/«^. 



^Jt: \n — I 



etc. 



etc. 



Substituting these expressions for X^ , X^ , ^3 , etc., in 
{b), we have Taylor*s formula* : 

f(x + h) = fx + fx h + f"x hV2 + f'"x liy|3 + . . . 

+ f«xhV|n + R, . . . . W 

in which fx represents what the given function becomes 
when h = o, f'x^ f"oc, etc., represent :he first, second. 



* Formula published in 171 5 by Dr. Brook Tavlor. 



D E VEL OF ME NTS. 1 73 

etc., derivatives of fx, and i? the sum of all of the terras 
after the {n ^ t)^^ 

The second member of {c) is also known as the develop- 
ment of the second state of a function of a single variable 

(§s)- 

Designating /(^ + ^) by jv', and/jc by j, we have 

which is another form of (c). 

To apply Taylor's formula, cause the variable with refer- 
ence to which the development is to be arranged^ to vanish. 
Differentiate the result and its derivatives in succession until 
one of the highest order desired is obtained^ and substitute 
them^ respectively, for their corresponding symbols in the 
formula. 

Thus, to develop (x -\- y)'*^, place 7 = 0, and differentiate 
x^y whence 

f(x) = x^, f\x) = mx'^-\ etc. 

y«(x) = m{m — 1) . . , , {m — n-\- i)x^"^. 

Substituting in (<:), we have 

1.2 

+ -^ A -^— ^ x"^-«y« + r: 

\n -^ ' 

125. Lagrange's Expression for the Remainder in 

Taylor's formula put x -\- h =^ X, whence h =■ X — x, 
giving 

fX=fx ^fx{X - x) -\-f'x{X - xy/2 + . . . 

+/M^-^)V)^+^. . . (i) 



174 DIFFERENTIAL CALCULUS, 

Assume R = F{X — x)" ^ VV ~^ ^j i^^ which F is an un- 
known function of X and x, which will make (i) exact for 
all values of x and X, giving 

fX -fx -fx{X -x)-... 

-r'x{X - xY/\n_ - F{X - xY-^y \n + i = o. . (2) 

Substitute z for x (except in F)^ and let Fz represent 
the result which in general will not be equal to zero, 
giving 

Fz =fX-fz -fz[X -z)- .., 

- rz{X - zYI\n_ - F(X - zY + V I;. + I . (3) 

From (2) and (3) we see that Fx = o, and from (3) we 
have FX =0. As s varies from x to Xy Fz increases and 
then decreases or the reverse, and F' z^ if continuous, must 
change its sign by vanishing for some value of z between x 
andX (§16, §63.) 

Differentiating (3) with respect to z and reducing, we 
have, since the terms with the exception of the last two 
cancel in pairs, 

F'z = - /" + ^^(AT - zYI\>l-\-F{X - zY/\n. 

Let x-\-(^n{X — x), in which 6*^ is a positive number less 
than unity, represent the value of z for which F'z = o. 
Then 

F=r^\x-i- 0^(X -> x)) = /" ^^\x + M), 



DEVELOPMENTS. 1/5 ^ 

and ^ R = f^^\x + e nh)h^''^^ /\ n + i *, 

in which o </9^<i. 

This expression for R enables us to determine the condi- 
tions of applicabihty of Taylor's formula. 

When as n-m-^oo , R is an infinitesimal, the formula gives 
a finite or convergent-infinite series for the function. 

This condition is fulfilled when as 7im-^co ^ fx and /"'^jc 
remain continuous between states corresponding to all values 
of X from any assumed value of x to x -\- h, for then 
/""^^{x-^-dnh) is always real and finite, and (§41) /^"+VU+i 
approaches zero as a limit. f 

With any assumed value of x which fulfils the above 
condition, h will frequently have limiting values. They are 
the numerically least positive and negative values of h that 
cause yir or any of its derivatives to become discontinuous 
when X -\- h \s substituted for x. 

If, for any assumed value of x, fx or any of its deriva- 
tives of a finite order, becomes imaginary or infinite, the 
corresponding limiting value of h must be zero, and the 
formula is inapplicable. It may, however, develop the func- 
tion for other values of x. 

To illustrate the use of R, take the example 

{x + yf^ = x^ + pix'^-'y -f '-^^i^-^ x^'-y -f . . . 

+ ^^"^ ~ ^^ • • , J^ ~ ^ '^ "^ x-y- + R, 

in which /'^jc = m{m — i) . . . . [m — n-\- i)x*''"^, 
7n(m — 1) . . . (m — n) y"'^^ 



and R = 



\n^i [x^ 6,yY 



-m+V 



* Known as Lagrange's expression for the remainder, 
t If, as «»»-»» , /^jcm-^sa /a requires evaluation, 



1/6 DIFFERENTIAL CALCULUS. 

When m is fractio7ial or negative, n may increase without 
limit, the series will have an unlimited number of terms, 
and m — n will become negative when n>m numerically, 
giving 

f^x = m(m — i) , . . (m — n-\- i)/x'^~'^. 

If ^ <o, the development fails when the exponent n — m 
is a fraction with an even denominator. 

If ^ = Ojfx ultimately becomes oo and the formula is 
inapplicable. 

If jc > o, we have the ratio of R to the (« + 2)th term 

equal to 

{xl{x + 6',;.))-'»+', 

which vanishes as n 3©-> oo ; and R, therefore, diminishes 
indefinitely when the successive terms in order likewise 
decrease. 

The ratio of the {n + i)th term to the nXh is 

m — n -\- \ y m/n — i + \/n y 

n X 1 x^ 

the limit of which, as n ^-> oo, is — (j/^v). 

Hence, when x is numerically greater than y the succes- 
sive terms in order will ultimately decrease indefinitely. 

In which case i? will be an infinitesimal; and we conclude 
that when 7n is fractional or negative, the binomial formula 
develops (^H-jf)"" for all positive values of x numerically 
greater than 7. 

EXAMPLES. 
I. Develop {x-\-yf'^. 

f{x) = |/^ f'{x) = 1/2 V^, f"{x) = - 1/4 \fj\ etc. 

(^ + J)'/^ = 4/^-1- y/2 i/x- //8 \/7' 4- R, 

which fails for ;tr = or < o. 



D E VEL OP MEN TS. 177 

2. Develop cos {x -\- y). 

f{x) = cos X, f\x) = — sin x, f"{x) = — cos x, 

f"\x) — sin X, etc. 
cos (x -\- y^ = cos X — y sin X — y^ cos x/2 -\- y^ sin ■^/|3_ 

+ y* cos x/|4_-l- i?, 
which is true for all values of x and^. 
Making x = o, we have 

cos ;>/ = I — y/2 +y/|4_— /l6_+ ^. 

3. sin {x -\-y) = sin x -]- y cos x — y"^ sin x/2 — y cos x/\$ 

-\-y* sin x/|4^+y cos x/\s_-j- R. 
Whence . sin ;/ = ;/ - y/|3 -|_y/|5__ //|7_+ ^. 

4. sin-l(jc+^) = sin-i;t: -j -} 



I + 2x' f 3x(3 4- 2x^) y I ^ 

t/(i - xo^ 11 \/{i - x'y li 

which fails when jr = i or > i numterically. 
For values of jr < 1 numerically, limiting value of y =^ i — x. 
Making x = o, we have 

sin-i;/ = -j,4-y/|3_4. 3y/j^+ ^. 

which is true for all values of x andy. 

Whence ta.n-^y = y — y^/3 -\-y^/s + ■^« 

6. Develop loga(x -f- y)' 

fx = logaX, f'x = Ma/x, f'x = — Ma/x"^, etC, 

/«x = (- iY-^Ma\ n - I /x^ Hence, 

iog.(. +^) = iog„. + i/„(z _ £, + ^ _ . . . +(_,)„-.£_)+,. 

In which i? = ± May^'^'^/in -f- i)(x -|- 6^^)'*+^ is an infinitesimal, 
as wm-^ 00 when x = or > j numerically. 



17S DIFFERENTIAL CALCULUS. 

\i X — Q>, the formula fails. If ;r = i, we have 

ioga(i -^y) = Ma{y-fl2 + ...-!-(- i)«-{rV«) 4- R, 

log (I +;^) = / - ;^V2 + y/3 - jV4 + ^. 
7. a^+jv = «^(i -^Xogay -\- \og^ay'^/2 + . . . + \og'>'ay»/\n) + R. 

R = a> ^ "^' log ^ a_y /[wj-j 
is an infinitesimal as n n-^ 00 , since log a is a constant, while 
[y/i^ ~{~ i) I '^"^ ^- Making ^ = o, we have 

a^ = I -\- log ay + log^ ay^/2 + . . . + log«fl!jj/Vt+ -^0. 

8. Develop (x^/^+/)^ 

The variables are \/x and y^. 

When the variables considered are not represented by the first 
powers of letters, substitute the first powers of other letters for the 
variables, develop the result, and resubstitute the variables for the 
auxiliary letters. 

Thus, placing x^^^ = r, and y"^ — s, we have 

(x'/' +/)' = (^ + ^)\ Ar) = r\ f\r) = Sr\ f"{r) = 20r^, etc. ; 
Hence, 
(y/2 -\-y^y = {r + sf ■= r^ -f sr*s + lor^^^^ + lorV + 5^^* -{- s^ 

= //^+ 5xv' + lo/zy-f IOX/+ s^^^y +y^'^' 

9. V^ + x+y -= V^T^+ -7=-- -7=== 4- R, 
which fails when x = — a or < — a. 

10. (x-a +yf^ = {x- af^ + six-af'y/2+is(x- a+Qyfy/S. 
x = a gives //' = i50'/y/V8, .'. B = 64/225; 

but the development would fail with more terms. 
\i X < a, the formula fails. 

11. tan {x -\-y) = tan x -j- sec' xy-{- 2 sec'* x tan jfj/y2 

4- 2 sec' Jf(i + 3 tan' j^)//|3,+ -ff. 

Inapplicable when x s tt/s. 



DEVELOPMENTS. 179 

1/ ix^ — I y^ 

14. sec-i ix -\-y) = sec-i X -| ; ^ h -^• 

15. cos'' {x — y)z=. cos'* ^ +JK sin 2x —y^ cos 2x — 2y^ sin 2j:/3 

+JJ/* cos IX / 2, + 2^ sin 2x/i5 + i?. 

16. 2(;f + >')' - 3(^ + J)' + I = ^'(2x - 3) + I 

+ 6(jf2 _ x)y -\- 2,{2x — i)y + 2JJ/3. 

t^;c— J t^x 34/-^* 9r-^ 274/ j^^" |3_ 

18. («;. + a/ff = a\x^ + 3^7/ + 3Vy + i//). 

19. log sin {x -\-y) = log sin x '-\-y cot x — y"^ cosec' x/2 

-\-y^ cos x/2) sin' ;i: -f- ■^. 

3 9 81 



21 






+ 

22. (- jt' + x)-^ = y-^ -\- 2XJ-3 + 3x2jj/-4 - 4;c3^-5 + R. 

23. sinh (;r +^) = sinh x{i +y/(2_+y/)4_+ i?) 

+ cosh^(;/4-y/|3_+^'). 

24. cosh (;c 4-j}') = cosh x{i -\-y'^/2 +y/|£+ R) 

+ sinh ^(^ +y/|l+ ^V|5_+ ^')- 

126. Stirling's Formula. — In Taylor's formula inter- 
change the symbols x and h, and place h =^ o, giving 

fic = fo + f'ox4-f"o^ + ...+ro-J' + R, . (a) 



I80 DIFFERENTIAL CALCULUS. 

which is Stirling's * formula for developing a function of a 
single variable into a series arraiiged according to the asceiid- 
ing powers of the variable^ with constant coefficients. 

/o, /'o, etc., represent what the given function and its 
successive derivatives respectively become when the vari« 
able vanishes. 

Placing /x = Uy we write 

-1 , du'\ d'u-] x' . . d^u~] x^ . ^ 

^ = 4 + ^Jo^+^ Jo 7 + • • • + ^ Jo |7 + ^- 

To apply the formula, differentiate the function and its 
derivatives in succession imtil one of the highest order desired 
is obtained. In the function and its derivatives make the vari' 
able equal to zero^ and substitute the results^ respectively ^ for 
their corresponding symbols in the fori7iula. 

Thus, develop (i + xY". 

f{x) = (i + ^)^ f\x) = m{i + x)^-\ . . . 

f'^ix) = m(m — i) . . . (m — n -\- i){i -\- x)^~\ 

/(o) = I, /'(o) = ^> /"(o) = ^(^ — i), etc., 

fn{o) = m(m — i) . . . (m — n -\- i). 
' Hence, 

(i + x)"^ = I + ^x + ^(^ — 1)0^/2 + . . . 

+ m(m — i) . . , {m — n -\- i)x'^/^n + i?. 

127. From § 125 we have, by interchanging x and h and 
making h = o, 

* This formula is generally known as Maclaurin's, but it was pub- 
lished by Stirling in 1717; and not by Maclaurin till 1742. It is a 
particular case of Taylor's formula which was published in 171 5. 



DE VEL CEMENTS. 1 8 1 

in which x^^'^ /\n +1 :^-^oas;2:^-^co (g 41). 

R is therefore an infinitesimal as n ^-> 00 , provided fx 
is continuous for all values of x from zero to the value 
assumed, \i f'^x'^^^ as /2^->co, i? must be evaluated. 

The limiting values of x in any case are the numerically 
least negative and positive values of x that cause /a^ or any 
of its derivatives to become discontinuous. 

It is important to note that if j^** o is imaginary or infinite 
for any value of ;?, the formula is inapplicable for all values 
of X. 

To illustrate the use of R, take the example in §126, 
whence 

R = -(---')^(---) (. + .„.)^-»->.«... 

JV/ien m is fractional or negative, and n is increased with- 
out limit, m — n will become negative, giving 

(i + xy-'' = 1/(1 + xy-^ ; also 

/-x = m{m- 1) . . . {m-n+ i)/(i + ^)— j«^«,= c^ ; 

and i?B-»co . o. The evaluation of R becomes necessary in 
this case. Being laborious it is omitted, but it will show 
that R is infinitesimal provided x < 1 numerically, and the 
series will be converging; otherwise not. 



1 82 DIFFERENTIAL CALCULUS:, 

EXAMPLES. 

.-. /(O) = O. 

.-. T\o) = Ma. 

.', /"(O) =-Ma. 

.'. f"'{0) = iMa, 

.-. /i^(0)= -\^Ma, 

etc. 

\n — \ Ma 
fn{x) = (- i)«-li^==^. /«(0) = (- l)«-l|«-j Ma, 



I. Develop loga 


(1+^). 


/[x) = 


]0ga (I + ^). 


fix) = 


Ma 


r(x) = 


Ma 

(i + ^r 


f"'{x) = 


2Ma 

(i + xf 


/iv(x) = 


\i_Ma 


(I + xf ' 


etc 


;. 



Hence, 



and 



l0ga(l+x)=Ma(x-'i+-^. . . ±^W^, 

\ 2 3 nj 



log(i+x)=x-^ + | _... ±1^+^?', 



I x^+'^ 

in which i?' = (— i)^ — ; -, — ; — ;, — ;^ — - is an infinitesimal under 

^ ' « + I (l + dnX)n+l 

the law that n increases without limit, provided x = or < i nu- 
merically. 

2. (I - ^2)1/2 = I _ x'^/2 - X4/8 - etc. + J?. 

3. sin ;r = X — x^/\3_ + ^y |5_ — x'/ 17_ + R, 

in which R = sin [(« + i)7r/2 + Q;^.r]-y^+^/| ^' -j- i is an infinitesi- 
mal. 

4. cos ^ = I - xy|2 + x*/\4_ - ^y|6_ + ^ 
in which R = cos ((« + i)7r/2 + e«^)jr«+y|?H-j. 



D E VEL OPMENTS. 1 8 3 

Since cos x '= a^sin ;r/^x, the development of cos j: may be 
obtained by differentiating that of sin x 

The radian measure of i' is 0.000291 — , for which value the develop- 
ments of sin \' and cos i' converge rapidly, giving their values with 
great accuracy. 

x'^ x^ 
5. a^ = i-4-log a .x-\- log"'* a f- • • • + log«« \- J?, 

in which /^ = c^"^^ log a x / \n 4- i diminishes as n increases 
without limit. 

Placing a = g, we have 



2 \n \n -\- 1 

in which x = 1 gives 

and X =x 4/— i gives 

^»'-. = i + xi/-i--- -1— + -+etc. 



(-lV^-.)+(.-|' + e..)y- 



= cos X -\- 4/— I sin ;c {a) 

Substituting — x for x, we have 

^-X -1 _ ^^g ^ _ ^_ J gjj^ j^ (J,') 



Hence, cos ^ = ( ^^^"^ + ^"^^"OA, 

sin X = ( ."'^'"1 - .-^"^-0/2 i/^ 



. . (0 



which are known as Euler's expressions for the sine and cosine. 
From {c) we obtain 

xV'~\ -xV~\ 2xV~l 

e — e e — I . . 

|/— itanx== — — = -;=- ~ "7^ • • • V") 



1 84 DIFFERENTIAL CALCULUS, 

In (a) and {b) put x = mx, then 

^mxV—i- , ./ • 

e = cos mx ± y— i sin mx. 



demx 

or, since e 



cos mx ± |/— I sin mx = (cos ;c ± |/— i sin ^)'«, 

which is De Moivre's formula. 

Expanding the second member by the binomial formula, and equat- 
ing separately the real and imaginary parts, we obtain, m being a 
^ positive integer, finite expansions for cos mx and sin mx in terms of 
cos X and sin x. 

Thus, m =3 gives 

cos sx ± |/— I sin 3J»r = cos^;c +• 3 cos'^JtrCi V'— i sin jf) 

— 3 cos jc sin^ ;c T 4/-- I sin' Jp. 

Hence, cos sx = cos^; 

sin 3;r = 3 C( 

In (c) place \/— i = /, giving 

cos X = (^^» 4- <^ ' ^0/2 , sin X = ((f^* — e-^^)/2t. 

From which, putting xi for x, and multiplying both members of the 
second by t^, we have 

cos xi = (<?^ -|" ^~^)/2. sin jfi = i(e^ — ^~^)/2. 

If /(— x) = — /(x), the development of /(x) will con- 
tain powers of x of an odd degree only. Such functions 
are called odd functions.* Entire functions all terms of 
which are of an odd degree with respect to the variable are 
examples, also sin x, tan x^ cot x, cosec x, sin~^^, and 
tan~-^;ic. 

If/(— x) =/(x), the development of /(x) will contain 
powers of x of an even degree only. Such functions are 



Rice and Johnson's Calculus, § 169. 



DE VEL OPMENTS. 1 85 

called even functions, of which cos x, sec x, vers x, and all 
entire functions containing powers of the variable of an 
even degree only, are examples. 

6. Develop sec x. 

f\x) = sec X tan x, ,., ^'(o) _ ^ 

f\x) = sec X (I + 2 tan^^), ... /"(q) = i. 

f"\x) = sec ^ tan jr (5 -f- 6 tan^^p), ... y-'/'(o) = o. 

p^{x) = sec ^ (5 + 28 tan^x + 24 tan*;»r), .-. /i'(o) = 5. 

etc- etc. 
Hence, sec x = i +^2/2 + 5vJ:*/24 + 6iJfy72o4-^. 

7. Develop cos^x. 

/'(•^) = 3 sin X (sxn^x — i), .*. /'(q) = o. 

/'V) = 3 cos X (3 sin*;r — i), .•. /'(o) =— 3. 

f"'{x) := 3 sin X (i — 3 sin^jc -f- 6 cos'x), .*. f"\o) = o. 

/'^^{x) = 3 cos X (1 — g sin'^x — 12 sin^x + 6 cos^jc), .*. /^"(o) = 21. 
etc. etc 

Hence, cos^^f = i — 3^:2/2 + yx*/S — etc. 

8. Develop (i + e^Y^, 

f\x) = «(i + ^^f - V, . \ f\o) = «2«-*. 

/"(x) = «(i + ^^)«-2 ^-[(i + ,^) + ^^(« - I)], 

.-. /"(o) = n(n+i)2''-\ 

3 r(i+^")' + («-i)(i + ^->- n 

L + 2(« - l)(l + ^^)^^ + ^2^(W - I) (« - 2) J 

... y-(o) = ;,2(« + 3)2«-^ 
etc. etc. 

Hence, 
(I + e-r = ."fi + ^ + »(^1+J]fl + "'(" + 3) f. + ,,,1. 

|_ 2 2^.2 2^ |3 ' J 



1 86 DIFFERENTIAL CALCULUS. 



9. Develop e^ ^'^ -^ with respect to sin '^x. 

^m sin- 1^ ^ j_^ ^ gj^_ 1 ^ _^ w2^sin-i;r)V2 + w3(sin-i;p)Vj3_+ i?. 

10. Develop <?»« sin" at vvith reference to x. 

The general rule may be applied, but the following method of 
Undetermined Coefficients* is perhaps simpler in this case. 

Place «?»« sin-i x^y^ then dy/dx = me^ sin-i jc/ 4/1 — ^2, 

and d^'y/dx^ = ni^ e^ sin-i ^/(i _ ^2) _j_ ;^^^w sin-i ^(i — x^f /^ 

Hence (i — x^)d'^y/dx'^ — x dy/dx = w'^j/ (i) 

Assume >/ = A o -{- A ix -{• A ^x^ -\- A%x'^ -\- . . . -\- AnX^ + R. 

Then ^/^;t: = ^, + 2^2;>: + . . . + «^«;tr«-l + R\ 

d'^y/dx"^ = 2^a + . . . + w(« ~ i)AnX»-^ + ^". 

Substituting in (i) and equating the coefficients of x« in the two 
members, we find 

^«+2= ^«(^2 +«')/(«+!)(« + 2) (2) 

from which ^2, ^3, ^4, etc., may be determined in order when A^ 
and A I are known. 

Ao = ^"^sin-l ^j^ = 1, ^^ — ^gmw^r^xl^^ _ ^aj^ _ ^^ 
Hence (2), Ai = m^/\2^ Aa = m(m^ + iVlS. etc., and 

Comparing this result with that of example 9, we have, by equating 
the coefficients of m, 

yZ o2y.5 '22r2„7 

11. sin-i. = .+|+^ + 3Ai + ^. 
Similarly, by equating the coefficients of m^, 

* Todhunter's Calculus. 



DE VEL OP ME NTS, 1 8/ 

'3.4 3-4-5.6 3.4. 5.6.7.8 
Equating the coefficients of m^, we have 

13. (sin-i^)3 = ^3 + 3.^1 + I_yi^5 + 3. . 5,^1 4. E_ _^ ^)|^' + ^° 

Dividing both members of example 12 by 2 and differentiating, we 
have 

sin-i^ , 2x^ . 2.4Jr* 2 . 4 . 6jr' , ^ 

\/i- x^ 3 "^3. 5 ^3-5.7 

From which, multiplying both members by i —x"^, we obtain 

/ 2x1/2 . , X^ 2 X^ 2 ^ X'' 

15. (i — x^) ' sm-i^f = X V- R^ 

^ ^ ■' ' 3 3 5 3 5 7^ 

in which, putting x = sin 9, we have 

sin^e 2 sin* 6 2 4 sin«e , ^ 

16. 9 cot = I h ^. 

3 3 5 3 5 7 

17. When the determination of the successive derivatives of 
a higher order is laborious, a simpler method of Undetermined 
Coefficients may be employed provided the development oi f'{x) 
is known. 

Thus, since sin-^ x is an odd function, which vanishes with x, we 
assume 

f{x) — sin-ix = Ax-\- Bx^ + Cx^ -j- Dx'^ + etc. 

Differentiating, we have 

/»=--r==^ = ^ + 3^^' + 5C^*+ 7^^* + etc. , (i) 

4/1 — or^ 

Developing 1/ |/i — •^'^> we have, provided \ar < i, 

^ 2'2.4'2.4.6' 

3.5.7...(2>^-i )£!l^^^^_ . . (,) 

' 2 . 4 . 6 . . . 2« 



88 DIFFERENTIAL CALCULUS, 



(i) and (2) are identical. Hence, 



and coefficient of ^ + i term = 



2.3 2.4.5 2.4.6.7 

3 5 . 7 . . . (2« — l) 



2.4.6.. . in{in ■\- i)' 



and sin-i x ■=■ xA h - • . 

^2.3 '2.4.52.4.6.7' 

, 3 ■ 5 • 7 . . • (2/^ - i)x2"+i ^ 

2 . 4. 6 . . . 2W(2« + l) "*" ' 

which is convergent for jf < i, since each term is less than the cor- 
responding term in the geometrical progression x -{■ x^ -\- x^ -\- etc. 

It I T I "J 

^ = sin-1 - = - H ^ + ^ h :ff. 

6 2 2 2.3.8 2.4. 5.32 

From which tt = 3.14159 . . . 

17-I. Develop tan~"^ x. 

Since tan~^.r is an odd function which vanishes with x, we assume 

f(x) — tan-i x—Ax^ Bx^ + Or^ + Dx'' + etc. 

Differentiating, we have 

f\x) = 1/(1 + ^2) = ^ -f- 35^2 + 5 Cr* + 7Z>x« + etc. . {a) 

Developing 1/(1 -|- ;>:''»), we have, provided jc < i, 

(l+;*:'^)-l = l-.r2 + ;»:*-x«+ etc (3) 

(a) and (3) are identical. Hence, 

A = \, B=-i/3, C=i/5, Z> = -i/7, etc. 

and tan-i JIT = .»: 1 h etc.+ (- i)** ■ 1- R, 

3 5 7 ' 2«+i ' ' 

in which ^ is an infinitesimal as « 3®-^ qo provided x = or <i 
numerically. 

\i x=i — tan (7r/4), we have 

;r/4 = I - 1/3 -f 1/5 - 1/7 + R, 



D E VEL OP ME NTS. 1 89 

To obtain a development which converges more rapidly, let 
<f) = tan- 1(1/5), a"d 6 = J0 ihen 6 = 4 tan"! (1/5). Hence, 

4 tan — 4 tan' 120 

tan e = ^ ^ ^ , 7^ = • 

I — 6 tan'' -\- tan* 119 

We also have 

tan (0 - 45°) = (tan - i)/(i + tan 0) = 1/239. 

Therefore 

— 45^ = tan-i (1/239), and 45° = — tan-i (1/239), 

or ;r/4 = 4 tan-i (1/5) — tan-i (1/239).* 

Developing tan-i (1/5) and tan-i (1/239), and substituting in 
above, 



Tt _ ll I I I \ 

~ \239 ~ 3(239)^ 5(239)^ ~ ^ ^- j. 

which converges rapidly. Seven terms of the first set and three in 
thesecond give it — 3- '41592653589793 • • -t 

,8. ^-f-+log(i+.) = 2.-f +4-^-^+^. 

19. ^ cos ;c = I -r ^ - ix^l\'h_ - 4^V|4_- 5-^V|5_+ ^. 

20. ^sin a; = I -I- ;c + x^/^ — xV8 - x^ / \t^ - JtrV240 + R, 

21. log {x + VTT^') = ^ - -^713 + 3'^y|5_+ R* 

22. ^*t2 = x2 H- jtr3 + xY|2 + ;cy|3 + R. 

23. (l + x)/(l — x) = I + 2X + 2X'-^ + 2Jf3 -f- ^. 

24. log -^^ = -J- _ -^-^ + -^-L-^ + ^. 

— ^ _ 

* Known as Machin's Formula. 
f Haddon's Differential Calculus, 



190 DIFFERENTIAL CALCULUS. 

[Suggestion. — Put x/{x — i) = i — z, and in development of (i — 2) 
put z — 1/(1 — x).^ 

25. (i + 2;c + 3x2)-V2 — \ — X ^ ix^ — 7^V4 + 3^72 + R. 

26. COS-l »r = 7r/2 — ^ — j:Y2 . 3 ~ 3Jr-y2 . 4 . 5 -j- -^. 

27. ^V(^^ — x) = ^2 _ ^4/2 — xy|3 + R. 

28. (i - ^2)-2 = I 4. 2x2 + 3^4 _^ 4^6 ^ ^^ 

29. ^Vcos >r= i+^4--^'^ + 2jtV3 + •^V2 + 3^Vio + ^, 

30. ^^sinx — I _^ ^2 ^ ;p4/3 _j_ y^>_ 



31. V I + 4-^ + isjc'-^ — \ A^ 2x -\- 4^2 -f- ^. 

32. e-V'^^ = I - lA' + i/2;t;* - i/6jr6 + i?. , 

33. g-x^ r^ I _ ^2 _|_ ^4/2 -f ^, 

34. (I + ^•2)V^ = I + 5-^73 + 5-^V9 - 5^V8i + R. 

35. /an-i^ ^ I _p ;^ _}_ ^2/2 _ x^/6 - 7^724 + R, 

36. sin2 X = x2 — xV3 + 2x73' .5 + ^. 

37. log sec X = x^/2 -{-x*/i2 -\- jrV45 + R. 

38. {a'^-{-dxy/^ = a-\- dx/2a - b'^x''/8a^ ^ 3d^x^/4Sa^ -\- R, 
39; ■ ^^ log {i+x) = x-i- xV2 + 2x3/|3_ + 9^71 5_ + ^. 

40. tan Jtr = X + -^Vs + 2xVi5 -f: i7-^V3i5 + '^• 

41. log (i -\- sin x) = X — x^/2 -j- x^/6 — xVi2 + ^' 

42. log (i + e^) = log 2 + x/2 + x''/2^ - x*/2^\4_ + R. 

43. (^^ + ^-^)« = 2«(i + «;t:2/2 + n(2n — 2)xYl4 ) + i?. 

44. cot X = i/x — x/3 — x^/f . 5 — 2x^/2^ . 5 . 7 + i?. (By method 
of Undetermined Coefficients, assuming cotx = Ax—^-\- 
Aix -{- A^'^ 4" etc.) 

45. tan" X = X* -{- 4x6/3 + 6xV5 + ^• 

46. By means of Taylor's and Stirling's formulas deduce the fol' 

lowing : 

sin (x ± j) = sin x cos y ± cos x sin y, 
cos (x ± jj') = cos X cosy =F sin x sin jj/. 

47. sinh X = {e"" - e-^)/2 = x + ^y 3_ + xy|5_ + R. 



DEVELO PMEN TS. 1 9 1 

48. cosh x = {e^-^ e-^)/2 = i + ^^2 + xy\A_ + R. 

49. cosh'^x = I + nx''/2 + ii{2n — 2)xy |4_ + ^. 

50. log {s'm\\x/x) = x'^/6 — xViSoH- A'. 

51. ia.nh-'^x = x-{-x^/^-{-x^/~,-\^R, 

52. sinh-l (x/a) = x/a — x^/2 . 3a^ + 3^5/2 . 4 . 5a^ + i?o 

53. Given j/^ — ^y -^ x = o; develop y in terms of the ascending 
powers of x. 

y = fx, /b = o or ± 4/3. 

3;/^^ dy/dx — '^dy/dx -f- i = o gives 

dy/dx\ = - i/(3j/^ - 3)]^j = 1/3, for;/ = o. 

2y{dyjdxf -\-y\d^y/dx'') - dy/dx"" = o gives 

d'y/dx-^]^ = - ty{dy/dxy/i3v' - 3)]o = O. 
(dyY . (dy\ dy , dv d-'y , J^y d^y 

. . d'^y/dx^]^ =2/27. 

Hence, _)/ = x/3 -\- x^/3* + ^« 

54. Given 2_;/^ — _:j/;c- — 2 = o; shovir that 

y = I -\- x/2 . 3 — x^/2^2>* + ^• 

55. Given jv'^ — S/y = 6x; show that 

y = 2 -{- X — x^/2'^2 4- x*/2 . 3 . 4 + i?. 

56. Given j^x — Sy — Sx = o; show that 

;/ = - ^ - xV8 - 3;cV26 + R. 

57. Given ^y^x — y — 4 — o; show that 

y= - 4-4^x- 3i4yx' + i^. 

58. Given y'^ — a'jj/ + '^J-^ — x^ = o; show that 



192 DIFFERENTIAL CALCULUS. 

59. Given sin = ;ir sin (a -|- 0); show that 

(p =z mt -\- SAVi a . X -\- sin 2a . x'^/2 + 2 sin « . (3 — 4 sin^ «)>^y|3_-f- 1^. 

^ ^ X e'' -\- 1 ^ 

60. Develop .* 

By Stirling's formula we write 

Since /vV =/(— ^), the development contains no power of x of an 
odd degree. 

Writing (^^ + i)/(^^ — i) = i + 2/(^^ — i), we have 

X ^^ + I X j; ^ jc^-^ 

/(^) = r .1^ — ; = r + ::^^ — ; = - + jr^- ^ + 



2<?^— I 2'^*— I 2' ^^— I* 

Differentiating, we have 
^^(/(^)+/'(x))=/'(^) + ^ + '-^^ (I) 

,-(/(^) + 2/'(x)+/'V))=/''(x) + '-3^i±^^ (2) 

e- (A^) + 3/'(^) + ^f'\^) + /'"(^) ) = f'iA + '— V^- (3) 

etc. etc. 

Making jv = o in (i), (3), etc., we find 

/(o)=:i, /'(o)-l/6, /iv(o)=._i/3o, /vi(o) = l/42, 

/viii(o) =. - 1/30, etc. 
Hence, 

7<?^"^~ 6'7 3^|4 42|6^ 3^15 ' 

* Edward's Differential Calculus. 



DE VEL OP MEN TS. 1 9 3 

or, placing 

^ = ^„ - = B,, -- = ^„ -- = A, etc., 
6 . 30 42 30 



Ar\ ^ /y«* -^^ •Y'*' 

2 






The coefficients represented by B^, B^, B^^ etc., are 
used in the higher branches of analysis, and are called 
Bernoulli's numberS: 

128. Extension of Taylor's Formula to functions of two 
or more sums of two variables each. 

Let u—f{x, 7), X and J being independent variables; and 
let it be required to develop /(x + /^,ji/-l-^), in which h and 
k are variable increments of x andjK respectively. 

If, in/(.r, j), X be increased by h, and/(:\^ -\-h,y) be de- 
veloped by Taylor's formula ; then if, in each term of the 
result, _y be increased by /^ and developed in a similar man- 
ner as a function oi y -\- k, we shall havey"(jt: -\- h, y -[- k) =■ 
sum of the latter developments. 

Otherwise, develop (p{t) = f{x -\- hf, y -\- kt) as a func- 
tion of /, by Stirling's formula, and in the result make 
/= I. 

<t>{f) = f(po -^ht,y-\- kt), .-. 0(0) = f(x, y) = u. 

In order to express conveniently the successive deriva- 
tives with respect to /, place x -\- ht =^ w, and j^ + /^/ = .?, 
giving 0(/) =/(«/, s). 

Hence (§ 102), 

d(t>(t) ^ a0(/) -dw dc()(t) 9£ 
dt 'dw dt 'ds df 



194 DIFFERENTIAL CALCULUS. 

'dw dx /* 9j- dy 



But 



. (Ex. 7, p. 71.} 



Also dw/dt = A, 



'ds/dt ~ k. 






^/90(/)\ ^ 9V(/) 9^ j 9W) ?£ ^ 5M^U 4- 5^^ 
^/V ^jt: / ^^ 'dw dt dx 9i- ^/ dx"^ dx dy 



k. 



Hence, 



^ '^ ^jt:' ^ dxdy ' dy^ 



etc. 



etc. 



0'^+^(/) = d''^'(P{t)h''+^/dx^+' + (^ + iW+'(p{t)h''k/dx^dy 
+ . . . + d''+'(p{t)k''^'/dy^^\ 
Therefore 

0'(o) = {du/dx)/i + (a?^/^jj;)^, 

,„, \ S'^^TS I S'^^ 7 7 t 9'^^72 

(°)=^^^+^<^^^^ + ^^. 



etc. 



etc. 



^^■<»-')=[^'--+<'+')?^'»+«- 






^ = 0ni? 



DEVELOPMENTS. IQS 

Substituting in Stirling's formula, and making / = i, we 
have 

/(x + /,^ + .) = . + |. + |. 

+ bL^'^" ^ ^«'-^' -^Z ^ ^'^■^ -^Z ^ -tv' J ^ {a) 
+ etc. etc. 



It should be noticed that 

0'^+'(/) is, in general, a function of ^ + ^/.and jv + /^/ ; 

0''+^(o) is the same function of x and j ; 

(p^'^'^(dnt) is the same function of ^ + hdj and_)^ + >^6'„/ • 

0^+^(6'^) is the same function of x + ^'w// and y -\- 0,ik. 

Therefore the remainder term in {a), which is equivalent 

to — ■ — 0''+'(6'^), is the same function of x -\- d^^h and 



ji-\- J 



y + dnk that — ; — (p^^^{6) is of x and y. Hence, the re- 

n -\- 1 



mainder term, denoted by i?, may be written 



y=>' 4-0/1 /& 



From § 125 and § 127 we see that formula (a) develops 
the given function provided that, as n increases without 
limit, 0"(o) is real and finite and R is an infinitesimal or. 



196 DIFFERENTIAL CALCULUS. 

what is equivalent, provided that u and all of its successive 
partial derivatives of the n^^ order are continuous between 
all states, corresponding to values of x and y from any- 
assumed values io X -\- h and y -\- k, under the same law. 

Having u — f{x, y, z), we may, in a manner analogous 
to above, deduce 

^/ r 7 I ZL I 7\ I 9^7 I 3^71 I 9^7 



2\_dx' ' df ' ^s^ 

\dx dy dxdz dy dz /J 

+ etc. etc. 

dx dy dz 



+ 



k^ + I 



Ax-x^Qnh 



y—y-\-Qnk 
Z-Z-\-Qnl 



(^) 



the remainder term being indicated by the symbolic form 
described in § 109. 

In a similar manner, a formula for the development of a 
function of any number of sums of two variables each may 
be deduced. 

129. Extension of Stirling's Formula. — In {a), § 128, 
put jc = o and J = o; then write x and y for h and k re- 
spectively, giving 

^=/(x,j)=/(o,o) + g. + |;,]^^^^ 



+ etc. 

d^ dy A{K^.Ky) 



, I ra , a i^^' 



BEVEL PM EN TS. 1 97 

which is a formula for the development of any function of 
two variables in which /(o, o), (du/d.x)o, (9W^^)o, etc., 
denote constants resulting from making x = o and/ - • o in 
u, du/dx, 'd'^u/dy^^ etc., respectively. 

The conditions of applicability, for any assumed values 
of X and y, are that u and all of its successive partial deriva- 
tives shall be continuous for all values of x andj' from o to 
those assumed. 

In a similar manner we may deduce from {h), § 128, and 
its extension, corresponding formulas for the development 
of any function of three or more variables. 

EXAMPLES. 

1. Develop {x + hy^iy + ky. 

u = /{x, y) = x»y^, ——■ = mx^~'^y'^, -—- = nx**^y'^-'^y 
ax ay 

-— ^ = m{m — iW'^-Sy^, -— ^ = mnx^'^-'^y^-'^, — ^ = n{n— i)x^yn-Z, 
dx^ ' '^ dxdy ^ dy^ ^ ' -^ 

etc. etc. etc. 

Substituting in fornnula («), § 128, 

{x + hy^iy + ky = x^y'*' + rnx'^^-'^y^'h + «x'«jj/«-i/^ 

4- m{m — i);ir««- V/iV2 + /?mx'«-ij)«-M/& 
+ n{n - i)x^y^-^k'^/2 -\- R. 

2. Develop (;c + /^)2[(a-i-jj/) + y^]3. 

u=f{x,y)^x\a+yf, '^^ = 2x{a + y)\ ^ = ^x^a -^ y)K 

g=.(.+.)3 |^=6.(.+,)i |-r-6.-^(.+v). 

9 « / , X 9 « 9 " 

,.>,■■ = I2(g + r). 1 ;=i2;r, 7-1-7^=12. 



198 DIFFERENTIAL CALCULUS. 

Hence, substituting in formula {a), § 128, 
(x + h)Ha + J + kf = x\a ^yf + 2x{a -^ yfh + zx\a -\-yYk 

+ 3(« +^)V^2^ + (ix{a -\-y)hk'' + ^tr^i^ 

3. Develop ?< = <?•* sin jj/. 

^. s _ f'du\ _ fd'^uX _ _ f'du\ _ 

\dx }q \dx'^Jo ' ' \dy JQ 

\dxdyjQ \dy^ la 

i^\=., f^Uo, f9!fl^=-r. 

Xdx'dyJQ \dxdyy \dy^ Id 

etc. etc. 

x^v v^ x^v xv^ 
Hence, <f^sin y = y ^ xy + -^ ~6"^6 6~'~^* 

130. Theorems of Lagrange and Laplace. 

Suppose y = z -\- x(p{y), (i) 

in which x and z are independent, and let it be required to 
develop /{y) according to the ascending powers of x with 
coefficients which are functions of z. 

Placing u =/{y), in which case u will also be a function 
of X and z, then developing by Stirling's formula, we have 

\ax /x=o I \dx /x=o 1.2 

+(93 f+j,,. . . . (,; 

in which the coefficients of the different powers of x are 

function<^ of z. In order to determine their values, we 

9// d''i^ , , , . 

transform ~— , ,-7, etc., before making x = o. 
ax cix^ 



DE VEL OPMENTS. 1 99 

Differentiating (i) with respect to x^ we obtain 

-dy/dx = 0(>-) + x\dct)i^y)l^y\{^y/dx\ 
whence 'dy/dx = 0(jJ^)/[i — ^90(^)/9y]. 

Differentiating (i) with respect to z, we have 

dy/dz = i-{-x[d(Piy)/dy]{dy/dz), 

giving dy/dz = i/[i — xd(piy)/dy]. 

Hence dy/dx = <P{y){dy/dz). 



Since 
we have 

Observing that 



82/ _ dii dy du _ du dy 
dx ~ dy dx' dz ~ dy dz* 



du/dx=cp{y){du/dz) (3) 



d_ 

dx 



(«-"!) =l-(*w|) 



dy dxdz ^"^^^Uzdx' ' ' ^4) 



and that u =/{y) gives 

du ^ d/jy) dy du ^ df(y) dy 
dz dy dz^ dx dy dx* 

we have, differentiating (3), 

dx' dxV^-^'dzl dxV^-^' dy dzl 



200 DIFFERENTIAL CALCULUS. 

Hence, by (3), ^ = 1(^(7)'^). 
Again, 

Hence, by (3), |^! = £(^W'^)- 
Similarly, from — = -^;^^0(;,) _j, 

we deduce ^-^. = _^0(_,) _j. 

which shows that the law of formation is general. 
^ = o gives JJ^ = ^, u=/(z), and -- = —^. 

XT 9^1 ^^ \9/W 



etc. etc. 



Substituting these expressions in (2), we have 
, x" 9«-' /'v^«3/(«)\ , „ 



which is called Lagrange's Theoretn. 



DEVELOPMENTS. 20I 

Suppose / — F{z + x(p{t)). 
Placing/ — z-{- x(p{t), we have 

/ = F(y), u =f(t) =/(i^(j)), and y = z + x<p{F{y)), 

to which Lagrange's theorem is applicable provided we 
write f{F) for /, and <P{F) in place of 0; therefore we 
have 

. I ^" 9""' r-7^77i- 9/(^(^)) 1 

+ • • • + \n_ dz"-^ \J>K.n^)) dz J 

+ ..., 

which is called Laplace's Theorem. 

Since the theorems of Lagrange and Laplace depend upon 
that of Stirling's, they hold only when x is small enough to 
make the developments convergent. 

EXAMPLES. 

I. Develop ;j/ = z -\- xe** 

In this case /( r) = y, f{z) — z, 

(p(y) = e^ , and (p{z) = /. 

Hence, from Lagrange's theorem, we obtain 



2 



13. ' ' |« 



202 DIFFERENTIAL CALCULUS. 

2. Given log^ = ;rv, develop^. 

We may write y = e=>^y, and putting xy^=:y\ we have y = xe^ \ 
which may be developed by making 2 = and y =y in example i« 
giving 

Replacing y by xy and dividing by x, we have 

3. Develop y = z -{- xy^. 

Here <J){y) =jj/«, 0(2) = 2«. 

Hence, 

z -\- xy^ = z -{- z^x -\- 2«s2m-i 1_ 3^(3„ _ i)^n~z j. j^^ 

4. Develop ^ = 2 + (f sin jj/. 

Here x = e, (p{y) = s\ny, 0(2) = sin«, 



1M = 



2 sm 2 cos z = sm 23;. 



|^(0W') = 



6 sin cos2 2 — 3 sin^ 2 = (3/4)(3 sin 32 — sin z), 

etc. etc. 

Hence, 

y = z -{- e sin z -\ sin 22 + -^(3 -siri 33 — sin 2) + i?, 

5. Having y = z -\- e sm y, develop sin y. 
Here /(jj/) = sin y, f{z) = (p{z) = sin z, 

'Q/{z)/dz = cos 2, jf = ,?, 
0(2) Q/{z)/dz = sin 2 cos 2 = sin 22/2. 



"^ V^^^^ ~^ / ^ ^^^^"' ^ ^°^ ^^ ^ ^3 ^^" 32 - sin z)/4. 



D E VEL OP MEN TS. 203 

Hence, 

sin jj' = sin z "1 sia 20 + yCs sin 32 — sin z) + R. 

6. Having jj/ = z + <f sin/, develop sin 2y. 

Here /(jj/) = sin 2y, f{z) = sin 2z, 0(2) = sin a, 

'^f{z)/dz = 2 cos 22, X =■ e. 

Hence, 

(p{z)'^f{z) / dz = sin 2 2 cos 22 = sin 32 — sin z. 

^ .?* 

sin 2;/ = sin 22 -|- (sin 32 — sin z)e -f- — (sin'' 2 2 cos 22) 1- R, 

7. Similarly, develop sin 3/. 

In this case f{y) — sin 3jj/, f{z) — sin 32, 0(2) = sin z, 
'^f{z)/dz = 3 cos 32, X = ^. 

Hence, 

d . ^' 
sin 3/ = sin 32 + <? sin 2 3 cos 32 + "t^^^"'' ^ 3 cos 32) f- i?. 

Qf3 2 

8. Similarly, develop cos J. 

Here /(>/) = cosj, /(z) = cos 2, 0(2) = sin 2, 

^/{z)/dz = — sin z, X = e. 
Hence, 

cos / = cos z — e sin'' 2 — 3 sin' 2 cos 2 1- R. 

9. Having «< = ;// + (? sin «, develop u, sin «, sin 22<, sin 3«, and 
cos u in terms of t and ^. By comparison with examples 4, 5. 6, 7, 
and 8 we have 

u = nt-\- (sin nt)e+ sin 2«^— ^ + (3 sin 3^/ — sin nt)-- + i?. 

<? <?" 

sin « = sin nt + sin 2«/ 1- (3 sin 3«/ — sin ni)— -\- R. 

2 o 

sin 2M = sin 2ni -\- (sin 3«^ — sin nt)e -\- R. 

sin 2^ = sin 3«^ + i?. 

g /J 

cos « = cos W/ — (l — cos 2/2/) (- (3 COS 3W/ -- 3 COS «/)— + ^. 

2 O 



204 DIFFERENTIAL CALCULUS, ^ 

lo. Kepler's Problem.* 
Having nt =: u — 6 s'm u, (i) 



_ / iH-e W/^_ u 

2 ~ \I - 6/ 



^^" T = \Tzn.) t^" 2 ' <2) 



f = a(i — e COS?/), ^3) 

find 9 and r in terms of /. 

First develop 6 in terms of ?/. In (2) put I 1 = m^ and since 

from {d), Ex. 5, § 127, 

we have — =z = — ^^ — — \ 

-y— 1 (^ + 1)^" ~^ — {m — i) 
Hence, e = i7=^' 

Placing — j — = A, and |/— i = ?", we have 
w -f- I 

I — A^»» 
Taking the logarithms of both members, 
Qi = ui-\- log (I — Xe-^i) - log (i — A^«0 

\g-ui _j g-iui _| ^-3«t + . . .1 

/ A2 A3 \ 

A'' A3 

= Ui + A((?«» — ^-«»") H (^2m» _ ^-2ki) _{ (^giui _ ^-3m/^ -f- . , . 

2 3 

A" . . A3 . 

= 2/z 4- 2X1 sin « + 2 — i sin 2w + 2 — i sin 32/ + . . . 
2 3 

* Price's Calculus, Vol. HI. pp. 561-564, 



DEVELOPMENTS, 20 5 

Hence, 

^ ■=u-\- i\X sin u-\- — sin 2u -\ sin 3m + . . .1, . (4) 



in which A. = 



3 
„^-^ ^ (i + e)V^-(i-6)V^ 

I _ (I _ e^)t/2 e , 



-F +. 



In (4) and (3) substitute for u, sin u, sin 2m, sin 3M, and cos u their 
respective values in terms of / from example 9, replace A by its value 
in terms of e, omit terms involving powers of e higher than the third, 
and we have 

6 = «/ + 26 sin nt ^ sin int -\ (13 sin 3«/ — 3 sin nt) +. . ., 

4 12 

Ce^ 3e' \ 

I — e cos «/ -| (i — cos2«/) — (cos3«/ — cos«/)4-. . . J, 

which are important equations in Astronomy, 



2g6 diI'FErential calculus. 



CHAPTER XL 

MAXIMUM AND MINIMUM STATES. 

FUNCTIONS OF A SINGLE VARIABLE. 

131. A Maximum state of a continuous function of a 
single variable is a state greater than adjacent states which 
precede or follow it. Thus, fx has a maximum state 
corresponding to x ^=- a, provided that as h vanishes we 
have ultimately and continuously /(^ > /(^ ± h). 

A maximum state is, therefore, a state through which, as 
the variable increases conti?2uously, the function changes 
from an increasing to a decreasing function, and its first 
differential coefficient changes its sign from plus to 77iinus 

(§63).^ . 

A Minimum state is one less than adjacent states which 
precede or follow it. Thus, fa is a minimum provided 
that, as h vanishes, we have ultimately and continuously 
fa<f{a±h). 

A minimum state is, therefore, a state through which, 
as the variable ificr eases continuously, the function changes 
from a decreasing to an increasing function, and its first 
differential coefficient changes its sign from minus to plus. 

A maximum is not necessarily the greatest, nor a mini- 
mum the least, state of a function. 

A function may have several maximum and minimum 
states. 



MAXIMUM AND MINIMUM STATES. 



207 



To illustrate, let ABCD-IJ be the graph of a func- 
tion (§ 20). 




The ordinates PA, RC, — TE, and fF/ represent maxima 
states of tlie function, and QB, — SB, — ZF, and — VH 
represent minima states. 

QB > IIV illustrates the fact that a minimum state may 
be greater than a maximum. 

The ordinate at ^represents a zero maximum, and the 
ordinate at y^ a zero minimum. 

The point /at which two branches of a curve terminate 
with separate tangents is called a salient point, and the 
points C and F at which two branches of a curve terminate 
with a common tangent are called cusps. 

132. A continuous function must have at least one 
maximum or minimum state between any two equal states; 
for if, in passing through any state, the function is increasing 
it must change to decreasing, and if decreasing it must 
change to increasing, at least once before it can again 
arrive at that state. The maximum ordinate PA, between 
the equal ordinates ML and M' L' , illustrates the principle. 

Similarly, it may be shown that a continuous function 
Jias at least owe minimum state between any two maxima, 



208 



D IFFEREN TE4 L CAL CUL US. 



and one or more maxima between any two minima. That 
is, as the variable increases, maxima and minima of a con- 
tinuous function occur alternately. 

I33» The general definition given for maxima and 
minima assumes that the function is continuous, and 
that as the variable increases adjacent states precede 
and follow those considered. Some exceptional cases arise 
which are illustrated in the following figure. 
B 






As X increases, the function represented by the ordinate 
of the curve ABC in passing through PA has no adjacent 
preceding states, fx does not change its sign, and is neither 
zero nor infinite; yet as PA is smaller than adjacent states 
it is generally considered a minimum. 

At Q the positive ordinate is an asymptote to both 
branches of the curve; and although the unlimited value of 
the ordinate does not represent a possible value of the 
function, yet fx changes its sign; therefore the ordinate 
QB \^ said to be an infinite maximum. 

At R the positive ordinate is an asymptote to one branch 



MAXIMUM AND MINIMUM STATES. 209 

of the curve, and the negative ordinate to the other 
fx does not change its sign , therefore neither of the 
ordinates ± RC^ respectively, is considered as a maximum 
or a minimum. 

E is called a terminating point, and the corresponding 
ordinate SE is generally considered as a maximum althougli 
fx does not change its sign. 

METHODS OF DETERMINING MAXIMA AND MINIMA. 

I34* Any particular state oi fx, sls fa, may be examined 
directly by determining whether, as k vanishes from any 
definite value, we have ultimately and continuously 

fa > f(a ± h) or fa < f{a ± h). 

Thus, let fx— c^{x — a)\ 

X = a gives fa = c, and f(a ± h) = c -\- /z^ 

Hence, /<3; <f{a -k h) 2iS h vanishes from any value, and 
fa — ^ is a minimum. 

Again, let fx = {x — i){x — 2)^ 

X = 2 gives /2 — o, and /(2 ± /;) = (i ± Ji)fi. 

Hence, /2 < /(2 ± /^) as /^ vanishes, and f2 — o is a ^ero 
minimum. 
^:=4/3 gives /(4/3) = 4/27, and /(4/3±/0 = ±/^'-/z' + 4/27. 

Hence, /(4/3) > /(4/3 ± ^0 ^s h vanishes, 
and /(4/3) = 4/27 is a maximum. 

Let fx — sin x, 
X — n/2 gives /{jt/i) = I, and f{7r/2 ± h) — sin (tt/z ± /.). 



2IO DIFFERENTIAL CALCULUS. 

Hence, /(^/s) > /(7r/2 ± h) as h vanishes, 
and f{'^/^) = I is maximum. 

Jt — o gives /o = o, and /(o ± /^) = sin (± ^). 

Hence, /o < /(o + /^), and /o > /(o — /^), as h vanishes; 
therefore /o — o is neither a maximum nor a minimum. 

I35» In general, maxima and minima are determined by 
finding those values of x corresponding to which, as the 
variable increases, f'x changes its sign. 

Assuming that x increases continuously, that/ir is continu- 
ous, and that every state considered has adjacent preceding 
and following states, a maxiiiium state is characterized by a 
change of sign from plus to minus in the first differential co- 
efficient, and a minimum state by a corresponding change from 
minus to plus. 

Conversely, if, in passing through fa, f'x changes fro?n plus 
to minus, fa is a maximum, and if f'x changes from minus to 
plus, fa is a minimum. 

f'x, if disconti?iuous, may change its sign by passing 
through a double value, one positive and the other negative, 
as illustrated by a salient point or by passing through infin- 
ity as illustrated by a cusp with common tangent perpendic- 
ular to the axis of X. 

f'x, if continuous, can change its sign only by passing 
through zero. 

A maximum or minimum oi fx corresponding to a salient 
point is an exceptional case distinguished by a double value 
for f'x. 

Hence, in general, values of x corresponding to which 
f'x changes its sign as x increases, are real roots of one or 
the other of the two equations 

fx — o, . . . (i) and /'^=co. ... (2) 



MAXIMUM AND MINIMUM STATES. 



211 



The figure, p. 207, illustrates the fact indicated by (i) and 
(2), that, in general, the tangent corresponding to a maxi- 
mum or minimum ordinate of a curve is parallel or per- 
pendicular to the axis of abscissas. 

fx does not necessarily change its sign as x passes 
through roots of (i) and (2), as may be seen in the cases 
represented by particular ordinates of the following curves. 

Y 




At the points A and B^ where the tangents are parallel to 
X,f'x = o; and at C and D, where the tangents are per- 
pendicular to X, fx = 00, but/':r does not change its sign 
as/{x) passes through the corresponding states. 

The points A, B, C, and D are called /<?/;//^ of inflexion. 

The real roots of (i) and (2) are, therefore, called critical 
values^ and the next step is to determine which of them 
correspond to states at which fx changes its sign as x in- 
creases^ and, therefore, correspond to maxima or minima of 
fx. 

The general method, for any critical value, as a^ is to 
determine whether, as h vanishes, f'{a — h) andy'(« -\- h) 
ultimately have and retain different signs. If so, a cor- 
responds to a maximum or a minimum, according as the 
sign of f(a — h) is plus or minus. 



212 DIFFERENTIAL CALCULUS. 



EXAMPLES. 

1. Let fx = b^-{x - a)V\ 

Thenf'x = 2/3(x — a)i/3 = oo gives the critical value a. 

As k vanishes, /'(a — /z) is negative and /'{a -\- h) is positive. 

Hence, /a = <5 is a minimum. 

2. fx = 6x -|- 3x^ — 4x^. 

f'x = 6(i -|- X -- 2x^) = o gives the critical values i and — 1/2. 
f\\ — h) =■ 6/z(3 — 2h), which is positive when k < 3/2. 
/'(i -f- /^) = 6/z(— 3 — 2h), which is negative when h > o. 

Hence, /"i = 5 is a maximum. 
f'{— 1/2 — h) is negative when h > o. 
/'{— 1/2 -f- -^) is positive when k < 2/3. 

Hence, /(— 1/2) = — 7/4 is a minimum. 

3. /r = a -f^ (x - 3)1/3. 

/'jf = i/3{x — b)^/^ = 00 gives x = b, 
f'{b "^ h) are both positive for all values of h. 
Hence, fib) = « is neither a maximum nor a minimum. 

4. /x = (x - i)v + 2)^: 

f'x = (x — i)^(x + 2)\nx + 5) = o gives 

X — \, X = — 2, X = — 5/7. 
/i = o is a minimum. 

/(— 2) = o is neither a maximum nor a minimum. 
/(— 5/7) = 124.93/77 is a maximum. 

5. fx = {x + 2)V(x - 3)«. 

f'x = (x + 2)'^(x — i3)/(x — 3)3 = o and 00 gives 

X = — 2, X = 13, X = 3. 
yi3 := 135/4 is a minimum. 
y3 — 00 is a maximum. 

6, fx — {a — xy/{a — 2x). f{a/^), min. 



MAXIMUM AND MINIMUM STATES. ^13 

-7. fx = — ■ . /(— ), max. / — , min. 

8. fx = xy^(2a — x)y^. /o, min. /(4«/3), max. 

136. The preceding method, or that indicated in § 134, 
must be employed for testing critical values from the equa- 
tion f'x = 00 ; but when/'^,/"^, /'"^, etc., are contmu- 
ous for values of x adjacent to critical values, those derived 
from the equation fx = o may be examined by another 
method. 

In Taylor's formula (§ 124) put x = a, and \/rite ± A 
for k; then, since /'<2 = o, we have 

+ (± ky + y^+^a ± tiji)/\n + I, (i) 

which form is exact for continuous values of /i from zero to 
certain limits, provided /a,/'a^ etc., to include /''+^^, are 
real and finite. 

In order thsit fa may be a maximum, we must have ulti- 
mately, as k vanishes, 

/a>f(a±/i); 

and fa a minimum requires, under the same law, 

/a</ia±/i). 

That is, fa a maximum requires that, as // vanishes, 

Aa±A)-/a, 

and therefore that the second member of (i) shall ulti- 
mately become and remain negative ; and/^ a minimum re- 
quires that the second member of (i) shall, under the same 
law, ultimately become and remain positive. 



214 '^ DIFFERENTIAL CALCULUS. 

As h vanishes, the sign of the second member of (i) will 
ultimately become and remain the same as that of H''f"al2- 
and since h^ /2 is always positive, 

fa a maximum requires/"^ < o, and 
fa a minimum requires /"<3; > o. 

In the exceptional case when f"a = o, the sign of the 
second member of (i) will, under the law, ultimately depend 
upon that of ± H^f"'a/W, which changes with that of }i. fa 
cannot, therefore, be a maximum or a minimum unless 
± Iif'"a/\T^ = o, which requires/"'^ = o. 

If also f "'a = o, the sign of the second member of (i) 
will, under the law, ultimately depend upon that of 
Ay^a/U; and since /^Vk is always positive, 

fa a maximum requires /^""^ < o, and 
fa a minimum requires /^^dJ > o. 

By continuing the same method of reasoning it may be 
shown that, ii f^a is the first derivative in order which does 
not reduce to o, fa is neither a maximum nor a minimum if 
72 is odd, and that it is a maximum or a minimum if n is even, 
according sisf^'a is negative or positive. Hence, we have 
the following rule: 

Having fa = o, substitute a for x in the successive deriva- 
tives of fx in order ^ until a result other than o is obtained. If 
the correspondiftg derivative is of an odd order ^ fa is neither a 
maximum nor a minimum j but if it is of an even order, fa is 
a maximum or a minimum according as the residt is negative 
or positive. If a result oo is obtained., the 7?iethod of § 135 
should be employed. 

The relations l)etween the corresponding states of /jc,/':^, 



MAXIMUM AND MINIMUM STATES. 



215 



and f'x^ in a case where ABCDE is the graph of fx, are 
shown graphically in the following figure.* 




EXAMPLES. 

Find the values of the variable which correspond to maxima 
or minima of the followijtg functions: 

1. fx =x^— 5Jc4 + 5;c=*+ I. 

f'x = 5x* — 2o;i:^ + I S^"^ = o gives the critical values o, I, 3. 

f"x = 20^^ — 6ox^ -\- 30X. 

/"o =0. f"\ — — 10. /"3 = 90. 
Hence, 

/i = 2, maximum, /3 = — 26, minimum. 



/'"x = 60^2 



:20j:-|- 30, 



/'"o = 30, and 



fo == I is neither a maximum nor a minimum. 
2. /x = jr^ — f^x"^ -\- 24X — 7. 

f'x = ^ix"^ — 6x-\-S) = o gives x= 2 or x = 4. 
/";c = 3(2;c — 6), .'. /" 2 < o, /"4>0. 
Therefore /2, maximum, f^, minimum. 



Calcul, par P. Haag, page 71. 



2l6 DIFFERENTIAL CALCULUS. 

^. fx = sin^x cos X. 

f X =3 sin^jT cos^x — sin'x = o, gives x = 60°, etc. 
f"x =r — 10 sin^x cos ;c -|- 6 sin ;r cos^;r. 
Since sin 60°= Vs/^* ^"d cos 60° = 1/2, 

y" 60° = — 3 '^3/2, .•. /6o° = 34/3/16, maximum, 

4. jf* — Sx^ 4- 22^2 — 24JC-+ 12. -^ = 3, min. 

:t = 2, max. 
x = J, min. 

5. ^^ — 4x + 9, X = 2, min. 

6. JfV3 + fl!^'^ — 3a'jp. X = a, min. 

:r = — 3d!, max, 

X = 3, min. 

X = 1, max. 

jf = P/2,a, min. 

jf = — ^V3a, max. 

X = — I, max. 
jr = I, min. 

^ = I, max. 
X = — I, min. 

^ = <f = 2.71828 . . . max. 

X = 7t/4, min. 
X = 57r/4, max. 

13. /x = ^^ +<?-•* + 2 cos X, 

y^x = ^^ — ^-^ — 2 sin X, f"x ■=e^-\- e-^ — 2 cos Jf, 
/'";c = ^^ — e-^ + 2 sin x, /i^ x = <f^ -f <f-^ + 2 cos x, 

f'o z=/"0=/"'0 = 0, /ivo = 4. 

Hence, /o = 4 is a minimum. 



7. 


^3 _ 5^2 + gx -p 10. 


8. 


3^2^8 _ 34^ _j_ ^6, 




.;c«-^+i 


9- 


x^-j-x-^l 




X 


10. 


1+x-' 


II. 


xV^. 


12. 


sec X -\- cosec jr 



14- 



I -^ X tan X 



jv = cos X, max. 



MAXIMUM AND MINIMUM STATES. 217 

137. The following principles frequently facilitate the 
determination of maxima and minima: 

If F\_f{x)\ is an increasing function of /(jc), F'\^f{a)\ is 
a maximum or a minimum of ^[/(^)] according as/(^) is 
a maximum or a minimum oi f[x). 

If i^[/(^)] is a decreasing function of /(^), F\^f{a)\ is a 
maximum or a minimum according as f{a) is a minimum 
or a maximum. Hence — 

1°. Cf{a)^ in which C is a positive constant, is a maxi- 
mum or a minimum of Cf{x) according as/(^) is a maximum 
or a minimum of /(^). 

2^. C + /W is a maximum or a minimum according as 
f{a) is a maximum or a minimum. 

3°. C — f{a) is a minimum or a maximum according as 
/(^) is a maximum or a minimum. 

4°. The base being greater than unity, log f{a) is a 
maximum or a minimum according as f{a) is a maximum 
or a minimum ; also any value of x that makes ^^'^^ a maxi- 
mum or a minimum makes /(jjc) a maximum or a minimum. 

5°. i/f{a) is a maximum or a minimum according as 
f{a) is a minimum or a maximum. 

6°. ;« being any positive odd integer, [/(^)]" is a maxi- 
mum or a minimum according as f{a) is a maximum or a 
minimum. 

;2 being any positive even integer, \_f{a)'Y is a maximum 
or a minimum according as f{a) positive is a maximum or 
a minimum or/(^) negative is a minimum or a maximum. 

[/(^)]'* may, however, be a maximum or a minimum 
when /(^) is neither. Thus, {a^ — x^)^ is a minimum, 
whereas (^^ — x^)a is neither a minimum nor a maximum. 

A radical sign which affects the entire function may 
therefore be omitted, provided critical values which corre- 



2l8 DIFFERENTIAL CALCULUS. 

spond to maximum and minimum states of the power only 
are rejected. 

Thus, having f{x) — ± \/ 2ax^ — x\ we write 

0(j,) = \_f{x)\ = 2ax' - x\ 

Then <P\x) = 4ax — ^x"^ = o gives x = o and 4a/^. 

(p(o) is a minimum and 0(4^/3) is a maximum of ^{x). 

But/'(^) = ± (4a — ^x)/2V2a -- X = o gives x = 4^/3 
only, and /(o) is neither a maximum nor a minimum of 
/(x). 

Similarly, {x — 2ay/{x^ — c^) is a minimum when ^ = 2^, 
but (^ — 2a)/ ^x^ — d^ is not. 

To illustrate the use of the foregoing principles ; 



Let f(x) = 1/(5 + log |/4^V - 2l^x'), 



By 5° we take 5 + log 1/4^ V — 2l>x\ 



By 2° and 4° we take \^4px^ — 2bx^. 
By 6° and 1° we take 2bx^ — x^ = (p(x). 

<p\x) = 4bx — ^x"^ = o gives jjc — o, ^ = 4V3« 
V"(o) = 4^, 0"(4V3) = - 4'^- 

Hence, 

/(o) is a maximum and /{4b/ t,) is a minimum. 

138. In certain cases it is not necessary to determine the 
second derivative. 

1°. In case of one critical value only, and it is known 
that the function has a maximum .state or that it has a 
minimum state. 



MAXIMUM AND MINIMUM STATES. 219 

2°. If a is the only critical value, /(^) is a maximum or a 
minimum, provided f{a) is greater than or less than both 
f(a ± h) for any assumed value of h. 

3°. When /'(a) is composed of two or more factors, one 
of which reduces to o, for x = a,/'\a) may be determined 
without using /"(jc). 

Thus, let /'(x) = tp{x)(p{x). 
Then /'\x) = iix)(p'(x) + (p(x)tp\x). 

Supposing that tp{a) = 0, we have 
/"(a) = 4>(<')>P'(a). 

Hence, to obtain f"{a), multiply the differential coefficient 
of that factor of f\x) which reduces to o by the other factors, 
and substitute a. 

To illustrate, let 

f{x) = (x- «) V. 

f'(pc) = 2x(x — a)(2x — a) =^ o gives 

X ^ o, X = a, X = a/2. 

y"(o) = 2(x — a)(2x — a)o = 2^^ indicating a minimum. 

f"(a) = 2jc(2ji; — cL)a = 2a^, indicating a minimum. 

f"(a/2) = /\x{x — a)^,^ = — d^, indicating a maximum. 

EXAMPLES. 

J^ind the values of the variable which correspond to maxima 
or minima of the following functions : 

(x -f- 3)^ X = o, min. 

' (jc + 2)* ' X ^ — 2, max. 

{y — \f J = I, min. 

Q/ -f- 1)=^* Jf' = 5, max. 



220 



DIFFERENTIAL CALCULUS. 



3, 1/^2^2 _ ^4. 



4. 



2r;rj[:' — TtX^ 



2.r — ^3. 



5. a*x — X 

6. aj^' — ;«:*, 



^ = o, min. of power. 
•«■ = ± <^/V'2, max., min. 
^ = o, min. 
^ = 4r/3, max. 

X = a/4/3, max. 
X = 3rt;/4, max. 



7. x-" - X 



5/2. 



8. 



X^ - 2^" 



2^" + 8 
9. x'^. 

10 (2^x< + a^bx)/a^. 
b 



^ 1 



{c - xf 



b a 



X = o, mm. 

X = 16/25, max. 

X = o, max. 
jf = 1. 19, min. 
X = i/e, min. 

X = — a/2, min. 

^ ya 

X — — Y= 3"^^» min. 

j/^ _|_ |/^ 

6 = tan-i |/3/dz, min. 



sin cos 

13. fx = x/2 — x^ sin {i/x)f2. 

fx — 1/2 + cos (i/x)/2 — X sin (i/x). 

1,1 i-i I .1 I 
1 cos — = cos' — X sin — = 2jr sm — . cos — . 

2 2 ;f 2X X 2X 2X 

Hence, fx = cos — . ( cos 2x sin 1 = o gives 

2X \ 2;»r 2x1 



M 





2X 


\ 2X 


2XJ 




X 


= 1/^, 


JT = 00. 


f\i/n) = 


- 4/^^ 


indicating a maximum. 


a-^x 






X ■= — a, min. 


{a - xf ■ 


X = ay max.. 


ab 


._./«= + ''■' 



X Va" -^ b"" - x"" 



MAXIMUM AND MINIMUM STATES. 22 



PROBLEMS. 

1. Divide a number a into two such parts that the product of the 
mS^ power of one and the w*^ power of the other shall be a maximum. 

fx = x^^{a — xY, f'x — x"^-\a — xy^-'^[?na — (w + n)x] = o 

gives X ^ o, X ^ a, x = ma /{in -\- n), 

/"ma/{m -]-«)=: — {m -{- n)c, indicating a maximum. 

2. Divide a number a into two such factors that the sum of their 
squares shall be a minimum. 

fx=:x'^-\- a'^/x^, X =^ ± ^a, minimum. 

3. Into how many equal parts must a number a be divided that 
their continued product may be a maximum? 

Let X = the number of equal parts, then 

fx — {a/xY, . *. log fx = x (log a — log x). 

f'x = fx{— I -j- log a — log x) = o gives x = a/e, 
f"x{a/e) = W^(— e/a), indicating a maximum. 

4. Let A be the hypothenuse of a right triangle; find the lengths of 
the other sides when the area is a maximum. 



Let X = one side, then \/h^ — x^ = the other. 



/x = area = x r/i'^ — x'^/2. f'x = o gives /^^ — ix'^ = o, 
whence x = hj V2, f'\h/ V2) — — ^/iK 

5. What fraction exceeds its «*^ power by the greatest number 
possible ? 

Let X = fraction, then fx=x — x^. 

f'x = I — «jc«-i = o gives X = 1/ \n. 

„( r-h-\ I — 

f \\/ \n)^^—n{n—\)l ««~^ , indicating a maximum. 

6. Of all isoperimetrical rectangles which has the greatest area ? 



222 DIFFERENTIAL CALCULUS. 

7. On the right line A c B joining 

the two lights A and B, find the point between the lights of least 
illumination. 
' Let c = number of miles from A 10 B. 

Let X = number of miles from A to required point. 

Let a = intensity of the light A at i mile from A, 

Let d = intensity of the light ^ at i mile from B. 

Then 

a b 

fx — -^-\-7—^ — <A — intensity of light at point required. 

c ya 



gives X 



|/a -f /\/b 



8. Find the relation between the radius of the base and the alti- 
tude of a cylinder, open at the top, which shall just hold a given 
quantity of water and have its surface a minimum. 

Let X = radius of base, and j = altitude. Then 

Ttyx"^ = volume = v. .'. y = v/itx'^, 

Ttx'^ -f- 2Ttxy = surface = /(x). 
Hence, 

fx = TtX^ + ITCXv/nx'^ = Ttx'^ 4- 2v/x. 

f X = 2TIX — 2vlx* = o gives x = '^vjn and y = '[/v/tc. 

/"( I^z^/tt) = 67r, indicating a minimum. 

g. Find the maximum rectangle that can be inscribed in a given 
circle. 

Take the origin at the centre, and let 2x and 2y respectively equal 
the sides of the required rectangle. Then xy = 1/4 area of rectangle. 

x^ -{-y^ = R"^, in which R represents the radius of the circle, gives 



y= ± V^'' 



Hence, fx = ±x\/R-^ - x"^- 

/'jf = ±(2^'^x — 4Jr^) = o gives x = o and x^^R/ \2. 
/"{r/ ^2) = — 4^^, indicating a maximum. 



MAXIMUM AND MINIMUM STATES. 



223 . . 



10. Determine the maxitrum rectangle which can be inscribed in 
a triangle of given base and altitude. 

11. Show that the difference between the sine and the versed sine 
is a maximum when the angle is 45°. 

12. The base and ihe vertical angle of a triangle being given, show 
that its area is a maximum when it is isosceles. 

13. Show that of all triangles of a given perimeter (a) and given 
base {b), the isosceles has the greatest area. 

14. Divide a triangle whose sides are a, b, and c, respectively, into 
two equal parts by the minimum right line. 




Ans. ^{c — a-^b){c-\-a — b)/2, 

15. Through a given point {a, b) draw 
the shortest straight line terminating in 
^and Y. 

Let 9 = angle required line CB makes 
with X. 
Then 
CB=CF-{-FB = a/ cos Q + ^sin 0, 

which is a minimum when Q tan-i \^b/a, giving CB ={a^^^ -{- ^^^Y'"^ 
Similarly, show that 

OB + C>C7is a minimum when = tan-i ^bja'y 

OB X OCis a minimum when = tan-i(V«); 

0B-\- OC-^ CB is a minimum when = tan-i ^ + ^^ ; 

a -f- ^lab 

OB X OC X C!i5 is a minimum when 

la tan^ — 5 tan'^ + a tan — 23 = o. 

16. Determine the maximum right cone which can be inscribed in 
a sphere whose radius is A'. 



224 



DIFFERENTIAL CALCULUS. 




Let X = AF, and y = PB. 

Then 

tcv^x 
vol. = z/ = ~ — , but y^ = 2Fx — x\ 

Therefore v = {2F7tx^ — 7tx^)/3. 
dv/dx = nx{^R — 3-^)/3 = o 
gives X = o, X = 4R/3. 

d^v/dx'^ = 7r(4/? - 6^)/3 = - 4^V3]^=4;?/3. 

17. Find the radius of a circle such that the segment corresponding 
to an arc of a given length shall be a maximum. 

Let 20! = length of arc, and r = radius. a. 

Draw CD bisecting the arc, then 

ZDCA = 6/2 = a/r, and = 2a/r. 

Segment = sector BCAD — aBCA 

= r'9/2 - r"" sin 6/2 

z=i ra — r^ sin {2a/r)/2, 

which is a maximum when r = 2a/7t, and 
the segment is a semi-circle. 

18. With a given perimeter find the radius which makes the cor- 
responding circular sector a maximum. Ans. radius = 1/4 perimeter. 

19. Find the maximum right cylinder which can be inscribed in a 
given right cone. 

Let VA = a, BA = b, AC = x, 
CD =y, CV = a — x. 

Hence, vol. cylinder = z/ = Tty^x, 
VA -.ABwVC: DC, .'. y= b{a - x)/a. 
Therefore v = nl)'^{a — xfx/a^. 
Omitting nb'^/a^, we have 

J{x) = a^x - 2ax'' + x^. 





MAXIMUM AND MINIAiUM STATES. 



22 C 



f\x) = a^ — ^ax -j- 3Jt^ = o gives x ■= a or (7/3. 
/"[a/'i) = — la, therefore v = /\TTaP/2-i is a maximum. 

20. Circumscribe the minimum isos- 
celes triangle about the parabola j^=4«x. 

Let X = C/' = ^ (9, 7 = PM, h = OD. 
Then 

^ _ A. 

BD—{h + x)y/2x = {h-\- x)\/ax/x, 

and area A 

= {k -{-xf 'i/ax/x, X = h/2, min. 

21. Determine the minimum right 
cone circumscribing a given sphere. 

Let X = alt. = AD, y = radius of base, 
R = radius of sphere. 

Then V = vol. of cone = ny^x/^. 





y\R:\ s/x'- -f / \x - R. 
From vi^hich y"^ = R'^x/{x — 2R), 

and V = 7rR^x^/3{x - 2R). 
X = 4R, min. 

\B 



22. Find the maximum parabola that 
can be cut from a given right cone. 
Let AC = a, AB —b, DC = x. 



Then AD = a —x, DE= Via — x)x. 
Also, a :x: : d : DG. .' . DG— bx/a. 



Parabola = \b \/ax''^ — x'^/^a, 
X = 3«/4, max. 



23. Find the maximum isosceles triangle inscribed in a given 
circle. 




226 



D IFFEREN TIA L CA L CUL US. 




]Let r — radius CA, AB — AE — x, BE = 2y. 
Then area A— u =^y ^ x^ — y^. 
Also, u — 



AB y^ AE y^ BE 



4r 



x'^y 



x\/4) 



And u 



i/Ar^- X' 



. X = ^4/3, max. 



2r ir 

24. Find the maximum cylmder that can be inscribed in a given 
prolate spheroid. 

Let Q.X — axis, and y — radius of 
base, of required cylinder. 
Then, vol. of cylinder 

= z/ = 2iiy'^x — 27txb'^{a'* — x'^)/a^, 
which is a maximum for x — (2/ 4/3. 

25. Find the minimum isosceles triangle circumscribing a given 
circle. 

Let y = radius CE, x — BF^ 

2y - AD. 

Then area A = xy. 

Similar triangles, BCE, BED, give 

y : y : ^ x'' -\- y^ : x - r. 





Hence, y - r\/x/{x - 2r), and 



area ^ - xr \/x/{x - 2r), 
D which is a minimum for x = 3^. 

26. " A boatman 3 mi from shore goes to a point 5 
mi. down the shore in the shortest lime. He rows 4 B_ 
mi. and walks 5 mi an hour, Where did he land? 

Let B be the boat 3 mi. from S, vvhich is 5 mi. 
horn the point /*. Let IV be the landing-place, and 

Then, number of hours 

r- i ^ 1/9+^74 -i- (5 - x)/s, 
which is a minimum for jc = 4. 



*Todhunters Difl. Calc, p. 213. 



MAXIMUM AND MINIMUM STATES. 



227 



27. Through a given point P within an angle 
BAC draw a right line so that the triangle 
lormed shall be a minimum. D 

Draw PD parallel to AC, and let AD = a. 

PD = h, AX - X. ,'. DX =: X — a. 
Then x — a\b \\ x : AR. 

AXR — X AR-sAw A J 2., which is a minimum for x — 2a. 




XP=PR. 




28. The volume of a cylinder being constant, find its form vyhen the 
surface is a mmimum. Ans. Altitude = diameter of base. 

29. Find the height of a light A above the 
straight line OB when its intensity at ^ is a 
maximum. 

Let a = intensity of light at i foot from the 
light. 

B OA - y, OB-b, I OB A - 6. 

The intensity varies directly as sin 9, and mversely as BA . 
Intensity at B = ay/{b^ -|- y^f^ , which is a maximum when 

y = d \/2 1 2. 

30, Having y =^ x tan a — x'^/^h cos'^ a . 1°. Find the maximum 
value of y. 2*. Considering/ = o and a as varying, find maximum 
value of X. 

1st. y — h sin- a, a maximum, x = h sin 2a. 



Ans. 



2d. X — ■ 2h, a maximum, a = 45' 



31. On the right line CC\ -- a joining the centres of two spheres 
(radii A, r) find point (rom which the maximum spherical surface Is 
visible. 




22S DIFFERENTIAL CALCULUS. 

Let CP = X. .'. PCx - a - X. 

Area zone ASHB = inR X HS. 

X- Rv. R: CS. .-. CS := R^x and HS-R- R^/x. 
Hence, zone ASBH = 2;rA'(A' - R''/x). 
Similarly, zone DEL — 2ni(r — r'^/{a — x)\. 

Visible surface --^ 27t\^R'' + r^ - (^R^x + rV(« - ^))], which is 
a maximum for x = aR^^'^/^R^^^^ + r^^% 

32. Find the path of a ray of light from 
a point /i in one medium to a point B in 
another medium, such that a minimum 
time will be required for light to pass 
from A to B; the velocity of light in the 
first medium being V, and in the second 
v. [Fermat's Problem.] 

It is assumed that the required path is 
in a plane through A and B perpendicu- 
lar to the plane separating the media. 

Let ACB be the required path. Through A, C, and B draw per- 
pendiculars to DE. 

Let a - AE, b = DB, d- DE, 

Then AC = «/cos 0, BC = b/cos (p' , 

CE = a tan 0, CD — b tan (p' . 

a tan (p -\- b tan 0' ■= d. .'. d<p'/d(p = — a cos^ <P'/b cos'' 0, 

h 
Time =. i, from A to B = 




V cos V cos 0' 

3 



a'/ _ \ F cos 0/ 1 F' cos 07 id_0'\ 

70 ~~ d'0 + d0' \d0)' ^^ ^^^' 



</0 

df 

d0 



sin 



sin 
F'" 



sin = T^sin 0', 



d0' 

- o gives 

sin _ 
sin 



V_ 



(a) 



for t a minimum. 



MAXIMUM AND MINIMUM STATES. 



229 



Equation {a) expresses an important law, which is known as Snell's 
law of refraction. 

33. To inscribe in a given sphere a right cone with a maximum 
convex surface. ^B 

Let R = radius AC, x = AP, 



y — V2RX - x^= TB, 
s = 27tTBAB/2 
— convex surface. 

Then AB = M^Rx, and 



= It V4R' 



2Rx^, 




which is a maximum for x = 4/^/3. D 

34. From two points A and B draw two right lines to a point F 

in a given right line OX, so that 
AF 4" BF shall be a minimum. 

Let X— OF, a = OC, b— CA, 
a' = OD, b' = DB. 
Ans. /LAFC-DFB, or = 0'. 
P D In some cases the general 

method ( §135) apparently fails when it is obvious that maxima and 
minima stales exist. 

35. Find the maximum and minimum distances from a given ex- 
ternal point to a given circumference. 





Let A be the point, r = radius, a 
Then 



FA = Vr^ + a-" 



2ax, 



= AO, OB =x. 

and dFA/dx = — 2a. 



It is obvious, however, that AM is a minimum and AN a maxi- 
mum. The method of § 135 depends upon the assumption that the 
variable increases continuously ; whereas x in the above expression 



230 DIFFERENTIAL CALCULUS, 

changes from increasing to decreasing, or vice versa, as PA passes 
through AM 2.n& AN. 

Let 6 = angle AOP. Then x = r cos 0, and 



PA = Vr^ -\- d^ — 2ar cos 
dPAjdb = 2ar sin 6=0. 



o, min. 

11. max. 



Otherwise PA may be expressed in terms of y and the problem 
solved. 

36. Find those conjugate diameters in an ellipse whose sum is a 
maximum or a minimum. 

Let X and y be any two semi-conjugate diameters, and let j=j;'-f->'' 
and a=^ + <5' = f^ Then [Anal. Geom.]y^+y2 = t^ 



. •. / = V^2 _ ^'2 and J = y + V^^ — x"^. 
.*. ds/dx' = I — x' I Vc^ — x"^ = o gives x'^ —c'^/i =y2. 

That is, equal conjugate diameters are those whose sum is a maxi- 
mum. 

Expressing x' and y in terms of the inclination of x to the trans- 
verse axis, denoted by 6, we have ds/dB = {ds/dx')(dx'/dQ). 

a and b are, respectively, maximum and minimum states of x\ 
giving dx' /dB — o, and therefore ds/dB = o. Hence the sum of the 
axes is a minimum. 

37. " A rectangular hall 80 feet long, 40 feet wide, and 12 feet high 
has a spider in one corner of the ceiling. How long will it take the 
spider to crawl to the opposite corner o n the floo^ if he crawls a foot 
in one second on the wall and two feet in a second on the floor?"* 

Ans. 55.4754 seconds, minimum. 

1 39* ^^ find the maximum and minimum distances from 
a given plafie curve to a given point in its plane. 

Let y = fx be the equation of any plane curve, (a, b) 
the coordinates of any point in its plane, and i? the dis- 
tance from [a, b) to the point {x\y') on the curve. 

* Problem proposed by Professor H. C. Whitaker in American 

Mathematical Monthly, Vol. L No. 8. 



MAXIMUM AND MINIMUM STATES. 23 1 

-If {x' ^y') move along the curve, R in general becomes a 
varying distance measured on the radius vector joining (^, b) 
with the moving point {x' ^y') and i?" =(a:' — of + (jv' — ^)^ 
We wish ,to find the maximum and minimum values of R. 

Placing the first derivative of {x' — ay -\- (y' — by equal 
to zero, we have 

^' _ ^ _l- (/ _ b){dy/dx') = o. . . . (1) 

The equation of the normal to y ^= fx at {x',y') is 
y — y' = — {dx'/dy')(x — x'),^i^S. Hence (i) expresses 
the condition that (x',y') is on the normal through {a, b). 

The required value of R is therefore estimated along the 
normal through {a, b), and is a maximum or a minimum 
according as the second derivative (dropping the primes), 

I + (Jy/dxY + (/ - i){dy/dx'), 

is negative or positive, and, in general, is neither a maxi- 
mum nor a minimum when the second deiivative reduces 
to zero (§ 136). 

As {a, b) may be any point upon any normal, we con- 
clude that the radial distance of each point of a normal 
from the curve is, in general, a maximum or a minimum 
when measured upon the normal. (See figure, page 232.) 

Thus, let BAM be a normal to the curve NMO at M. 
^With A and B as centres, and with the radii AM a.nd BM 
respectively, describe the circumferences rMr and RMR. 
The figure shows that the radial distance of A from NMO 
is a minimum when measured upon the normal AM, and 
that the corresponding distance of the point ^ is a maxi- 
mum. This is evident from the fact that the circumference 
vMr^ in the vicinity of and on both sides of M, lies within 
the curve NMO, while the corresponding part of the cir- 
cumference RMR lies without. 



232 



DIFFERENTIAL CALCULUS. 



Consider the point [a^ b) to move upon the normal, and 
let yoc^yj be its variable coordinates. When the normal 




distance of \Xyy ) from the curve is neither a maximum nor 
a minimum, we have 

I + {dy/dxY + (y --y){d-^y/dx') = o, 

whence by combination with (i) we obtain 



dx' 



\ dx^ jdx ' dx 



(2) 



for the coordinates of a point on the normal whose distance 
from the curve measured along the normal is, in general, 
neither a maximum nor a minimum. Representing this dis- 
tance by p, we have 

p^ = {x-Tf + {y-y)\ 

which combined with (2) gives 



p= 1 + 



dx'' 



'■ l<£l 

I dx' 



(3) 



MAXIMUM AND MINIMUM STATES. 233 

In the figure, page 232, \x^y ) lies somewhere between A 
and B. It separates those points of the normal each of 
which has a minimum radial distance from the curve lying 
on the normal, from those points of the normal each of 
which has a corresponding maximum distance on the same 
line. 

It is important to observe that a circumference described 
with \x, J' ) as a centre and with a radius equal to p will, 
in general, intersect the curve NMO at M. 

This circle is important in the discussion of curves, and 
equations (2) and (3) will be referred to hereafter. 

IMPLICIT FUNCTIONS. 

140. Having J given as an implicit function of x^ by an 
equation f{^x^y) = o not readily solved with respect toj, 
we may differentiate as indicated in § no and obtain an 
expression for dy/dx. Placing it equal to o, we may com- 
bine the resulting equation with the given, and find critical 
values of x. 

Otherwise, let u=f{x^y) = o (i) 

Then (i), (§111), 

du/dx = du/dx -f (di^/dy) (dy/dx) = o. . (2) 

Maxima and minima values of y in general require 
dy/dx = o. Hence, 

^ —f(^,y) — o, combined with 'du/dx = o, \ 

gives critical values of x. 
Eq. (6) {§ in) gives 

d'^y/dx^ = - (dW^^l/idu/dy), 

which, if not zero or infinity, is positive for a minimum and 
negative for a maximum oi y. 



234 DIFFERENTIAL CALCULUS. 

Having y — fZj z = cpx, 

dy/dx = (dy/dz) X (dz/dx) = o, 
will give critical values of x. 

EXAMPLES. 
I. u = x^ -\- y^ — 2^^^ = o« 

du/dx = sx^ — Sa"^ = o. .'. X = ± a. 
Substituting in given equation, we havejj/ = ± a ^2. 
'd^ujdx^ = tx, 'du/'dy = s^y". 

^'Vn -6a 3/-. 

:74 \x=a = , 3/- > • • ji' = « y2 IS a maximum. 



^vn 6a 8/—. . . 

— ^n, . . jj/ = — a y 2 IS a mmimum. 



'1 - - — 

2. Jf^ — saxy -j-jj/^ = o. jr = o, jj/ = o, is a minimum. 

X = a 4/2, y — a I/4, is a maximum 



3. jc" -j-^'' — 23xy — a' = o. 
a3 



^ = , IS a maximum. 



4. 4XJJ' — ;j/4 — jc* = 2. X = ± I, >' = ± I, no max. or min. 

5. jv'^ — 3 = — 2Jr(jj/x + 2). X = — 1/2, J = 2, a maximum. 

6. jj/ = Tt^zjuy z r= {k"^ -\- x'^)/x, X z= k makes ^ a minimum. 

141. Having v given as an implicit function of x^ by two 
equations v = (p[x^y) and ti ^= f{x,y) = o, from which _y 
is not readily eliminated, we may proceed as follows : 

dv _'dv dv dy 
dx dx 'dy dx ' 



dy 'du fdu ( \ ro X 



dx dxl 'dy 



MAXIMUM AND MINIMUM STATES. 235 

' dx ^ dx dy dx I Qy ' 

dv . dv 9^/ dv du , . 

and ^=0 gives -— _---=o, . . (i) 

which combined with u = /(x, y) = o gives critical values 
of X. The sign of the corresponding value of d'^v/dx^ will, 
in general, determine whether z/ is a maximum or a minimum. 

EXAMPLES. 

I. 2/ = x«+/, (;»: - «)' -\-{y- bf - c^ = 0. 

dv/dx — 2x, dv/dy = 2y, 

du/dx = 2{x — a), du/dy — 2{y — 5). 

Substituting in (i), we have ay z=dx; which combined with m = o 
gives 



X = a ±ac/ Vd" -h b"". 

The positive sign gives a maximum, and the negative a minimum, 
for V. 

2. Find the points in the circumference of a given circle which are 
at a maximum or minimum distance from a given point. 

3 Given the four sides of a quadrilateral, to find when its area is a 
maximum. 

Let a, b, c, d be the lengths of the sides, (p the angle between a and 
b, rj) that between c and d. 

Then area r=. v = ab sin 0/2 -|- cd sin ^/2, 

and a" _|_ ^2 _ 2^(5 cos (p = c^ ^ d"^ — 2cd cos ^, 

each member being the square of the same diagonal. 

dv ab , dv cd 

-— = — cos 0, — — = — cos tp, 

d(p 2 ^' a^ 2 

-7- = 2«<5 sm 0, — = — 2cd sm z^. 

^0 d^i^ 



236 



DIFFERENTIA L CA L CUL US. 



Substituting in (i), we have 

tan (p — — tan ^. . ". (j) — 180° — ^. 
That is, the quadrilateral is inscribable in a circle. 
dv ab ^ , cd , d^ 

— - = cos (p \ cos Tp—- — O, 

d(p 2 '2 d(f) 

ab sin (f)=^ cd sin '^{dip/d(p) ; 
from which dTp/d(p = ab sin cp/cd sin ip. 

Substituting in above, we have 

dv/d<p = ff^ sin {<p -\- ip)/2 sin ip. 
d'^v _ ab cos {(p -(- ip) 



d(p'' 



d'^v' 



— ab i , ab\ . ,. 

— : — -, U H ;l> indicating 

2 sin ip\ cdr 



a maximum. 

, . , 2 sin ■ib\ ' ccti " 

142. Having w given as an implicit function of x, by- 
three equations 
w = F{x, y, z), V = (p{x, y, z) = o, and u = /{x, y, z) = o, 

and placing dw/dx = o for a maximum or a minimum, we 
write 



dw 9w 9w dy 'dw dz 

dx dx "dy dx 'dz dx 



dv 
dx 


dv dv dy dv dz 
~~dx '^ dy dx'^dz ^ ~ ° 


du 

dx 


du du- dy du dz 
dx dy dx dz dx 



(l) 



Eliminating dy/dx and dz/dx, we have a single equation 
which, combined with 7V = F{x,y, z), v = o, and u — o, 
gives critical values of x and the corresponding values oi y^ 
z, and w. 

By differentiating equations (i), and eliminating 

dy/dx, dz/dx, d'^y/dx', d'^z/dx'^, 
^n expression for d'^w/dx^ may be determined. 



MAXIMUM AND MINIMUM STATES. 237 

Example. * A Norman window consists of a rectangle 
surmounted by a semicircle. With a given perimeter, find 
the height and width of the window when its area is a 
maximum. 

Let y = height, 2X — width, w = area, J^ = perimeter. 

Ttx"^ 
Then w = j- 2xy, v = 2{x -\-y) ■\- nx — P =-0. 

dw/dx = Ttx + 2 V + 2xdy/dx = o, ) 

[.. . (i) 

2 -{- TV -\- 2dy/dx = o. ) 
Eliminating dy/dx^ and combining result with z^ = o, 

Differentiating equation (i), we have 

d'^w/dx' = TT -{- 4dy/dx + 2xdy/dx^f 
2d^y/dx^ — o. 
Hence, d^w/dx'^j^^y = — tt — 4, indicating a maximum. 

FUNCTIONS OF TWO OR MORE VARIABLES. 

143. A Maximum state of a continuous function of two 
independent variables is one greater than any adjacent 
state. Thus, z=/{x,y) is a maximum corresponding to 
X = a, y = d, provided as /i and k vanish from any values, 
we have ultimately and continuously /(^, b)> fia^h^ d±k). 

A maximum state is, therefore, one through which, as 
either or both variables increase continuously, the function 
changes from an increasing to a decreasing function, and its 
partial differential coefficients of the first order change their 
signs from plus to minus. (§71.) 

*Todhunter*s Diff. Calc, p. 214. 



238 DIFFERENTIAL CALCULUS. 

A Minimum state is one less than adjacent states. Thus, 
/{a, b) is a minimum, provided as h and k vanish from any 
values, we have ultimately and continuously 

f{a, b) <f(a±h,b± k). 

A minimum state is, therefore, one through which, as 
either or both variables increase continuously^ the function 
changes from a decreasing to an increasing function, and iis 
partial differential coefficie7its of the first order change their 
signs from ?ninus to plus. (§ 71.) 

Any particular state of z =/(x, jf), as/(^, ^), may be ex- 
amined directly by determining whether as h and k vanish 
from any assumed values, we have ultimately 

f{a,b)> f{a±h,b±k), or f{^a, b)<f(a±h, b±k). 

144. In general, however, maxima and minima are deter- 
mined by testing those values of the variables correspond- 
ing to which, as the variables increase, both partial 
derivatives of the first order change their signs from plus 
to minus, or minus to plus. 

Sets of roots of the equations 



'dz/dx = o, 'dz/dy = 0, . . . . (i) 

( dz/dx — 00 , dz/dy = co 
'Exceptional 

. cases. I 1 S"^/-^^ = "' ^'/'^^ = ~ (s) 



(3) 

[ dz/dy — 00 , 'dz/dx =: o, . . . . (4) 



are, therefore, critical, and may be tested as indicated in 

§ 143- 

A maximum or minimum state of a function of two va- 
riables is illustrated by an ordinate of a surface which is 
either greater or less than all adjacent ordinates. The 



MA XIMUM A ND MINIM UM S TA TES. 239 

conditions dz/dx = o and dz/dy — o indicate, in general, 
that the corresponding tangent plane is parallel to XY. 

145. Lagrange^s Condition.— When the successive partial 
differential coefficients to include those of the n + 1 order 
are real and finite for a set of critical values, as {a, b), de- 
rived from equations (i), § 144, a condition for a maximum 
or a minimum may be deduced from (^), § 128. 

Placing 

m =A (^ =B {^y\ ^C 

\dx^l(a,b) ' \dxdy)^a,V) ' \dy''}(a,b) * 

we have 

/{a ±h,b±k) -f{a, b) = {Ah' ± 2Bhk + C/^'^)/2 + R. (i) 

In general, as h and >^ vanish, the sign of the second mem- 
ber will ultimately depend upon that of A?^ ± 2Bhk -\- Ck^ ^ 
and when A ^ o \\. may be written 

\i^Ah ± Bky + {AC - B')k'']/A. 

A maximum or a. minimum state will then require that the 
sign of this expression shall ultimately become fixed and 
remain so, while h and k change their signs by passing 
through zero. 

If {AC — B^) < o, the numerator of the above expression 
will be positive when k = o, and it will be negative for 
values of /i and ^ other than zero which make A/i ± Bk =0. 
Hence, a condition for a maximum or a minimum state is 

{AC- B')> o, or AC> B'; 
that is, p^ Xp:] >(-f^)l ...(.) 



240 DIFFERENTIAL CALCULUS 

This condition being satisfied, /(^, b) is a maximum or a 
minimum according as A and C are both negative or both 
positive. 

If AC <. B^y there is neither a maximum nor a minimum. 

If ^i = o and B :^ o, we have the same result, for the 
sign of 2Bhk + Ck^^ in the second member of eq (i), varies 
for a fixed value oi k d.-s, h passes through the value 
- Cki'2B. 

li A = B =^ o, then 

AC - B' =^ o, and A/i' ± 2BM + Ck' = Ck\ 

which vanishes when k ■= o, for all values of h leaving 

the question in doubt. The same result follows when 

A ^ B =: C ^ o. 

EXAMPLES. 

Find sets of values of the variables which correspond to 
maxima or minima of the following functions : 

1. z = x^ -\- xy -\- y^ -\- a^/x -\- a^/y. 

'dz/dx = 2x -]- y — a^/x"^ = o, 'dz/dy = 2y -\- x — a^/y^ = O. 
From which we obtain x— y — aj 1/3, 
8V^-^' = 2 + 2aVx^ a^V^r' = 2 + 2aVy, 'd^z/dxdy = I. 

In which substituting values of x and y, condition (2) is satisfied and 
« is a minimum. 

2.2 = cos X cos a -{- sin X sin a cos (y — f3). 

^z/dx = — sin X cos a -\- cos x sin a cos (y — /3) = o, 
C)z/dy = — sin a sin x sin (jj/ — ^) = o, 
give X = a, y — ft- 

f9!iU-,, (9!i) = _ sin'., f^)=o. 

V O'^^ / (a, ^) \ ^'^ / ia, /3) V^ ^1'/ (a, ^) 



MAXIMUM AND MINIMUM STATES, 24I 

Hence 3 is a maximum. 



X = a/2. 



\y 


= c^/3, 


IX. 




X 


= y = a, : 


min. 




X 


= y = o, 


max. 




X 


= y= ± 


i, min. 




X 


= ± W3 


>y='f 


il/3, min, 


X 

y 


= 2', "^^^• 






X 


= ± V2, 






y 


= TV2, 


min. 




X 


= 0=y, 


max. 




X 

y 


~ i' max, 






X 


= y = 0, 


min. 





3. xy^{a — X — y). 

4. x^ -\- y^ — 2>axy. 

5 . x^ -\- yi — x^ -\- xy ^ y^. 



6. jf^y(6 — X ^y\ \ 

7. X* -\- y^ — 2x^ -j- 4xy — 2y^. ■< 

8. {2ax - x"") {2by - y""). ] 

9. g-'^^-y\ax'^ + by""). 

X =■ o, y ^=- ± \, a <. b, max. 
X =■ ± I, J = o, ^7 > b, max. 

10. sin X -\-s\ny -\- cos {x -\- y)- x = y = 37r/2, min, 

X = y = 7t/6, max. 

11. Divide a number a into three parts, such that the ni^^ power of 
ihe first, by the «"* power of the second, by the r^^ power of the third 
shall be a maximum. 

Ans. ma, na, and ra, each divided by (w + « + '')• 

12. Find the minimum distance from a given point to a given plane. 

13. The volume of a rectangular parallelopipedon being given, find 
its edges when the surface is a minimum. Each edge = l^vol. 

14. An open tank to contain a given volume of water is to be con- 
structed in the form of a rectangular parallelopipedon. Determine its 
edges so that the surface to be limed shall be a minimum. 

Each edge of base = i/2 vol. ; altitude = ^^2 vol. /2. 

15. Determine the maximum rectangular parallelopipedon which 
can be inscribed in a given sphere. Each edge = 2i?/ 1/3. 

16. Of all isoperimetrical plane triangles which has the maxi- 
mum area ? 



242 DIFFERENIIAL CALCULUS, 

146. Functions of Three Variables.— Let 2/ =/(^,j, s). 
Reasoning as in the preceding cases, it may be shown that 
sets of roots of the equations 

'dtc/dx = o, 'du/dy ■= o, 'du/dz = 0, . . (i) 

and du/dx = 00, du/dy = 00 , du/dz = 00, . . (2) 

are critical. 

Denoting a set of critical values from (i) by a, d, and r, 
we have, {d), § 128, 

/{a±/i,d±J^,c± I) -/{a, b, c) = [A A" + Bk' + CP]/2 

±Dhk±Ehl±Fkl-\-R, 

in which A^ B^ C, Z>, E^ F, represent the values of 

a^ 3^ 8^ _a^ _a^ ^ 

dx''' dy" dz'' dxdy' dxdz' dy dz' 

respectively, when x =^ a, y =^ b, z ^^ c. 

In order that/(^, ^, c) may be a maximum or a minimum, 

^{AH' + ^/^^ + O") ± Dhk ± Ehl ± Fhl, 

if not zero, should be either always negative or always posi- 
tive, as h^ k, and / vary through zero between certain positive 
and negative limits. 

147. Functions of n Variables. — By extending the above 
method of reasoning, it maybe shown that the sets of roots 
of the equations formed by placing the partial derivatives 
of the first order separately equal to zero are critical. 
Each set of critical values when substituted in the corre- 
sponding expansion should render the quadratic function of 
h^ k, /, etc., always negative for a maximum, or always posi- 
tive for a minimum, as h^ k^ /, etc., vary through zero 
between certain limits. 



PART III. 

GEOMETRIC APPLICATIONS- 



CHAPTER XIL 

TANGENTS AND NORMALS. 
RECTANGULAR COORDINATES. 

148. Equations of a Tangent and NormaL— The equa- 
tion of a straight line passing through (jc', y') on the curve 
y =^ fx is (Anal. Geom.) y — y' = 7n{x — x'). 

Placing ifi —f'x' — dy / dx\ we have for the tangent line 
at(y,/)(§7i) 

y-y'^-Wldx'){x-x') (i) 

and for the corresponding normal 

y -/ = - {dxldy'){x - x'). ... (2) 

Thus, having y = ^x^ then f'x = 9/27, /'4 = 3/4. 
Hence, ^^ — 6 = (3/4) (x — 4) is the tangent, and 

y — 6= — (4/3)(j\: — 4) is the normal at (4., 6). 

If the equation of a line is in the form u — <p[x^y) = o, 
we have (§ in) 

/'x' = dy'/dx' = - {du/dx')/{du/dy). 

243 



244 DIFFERENTIAL CALCULUS. 

Substituting in above, we liave 
{bu/dy'){y — y) = — i(du/dx'){x — x') for the tangent, 
{^u/dx')[y — /) =: {^^n,/dy'){x — x) for the normal. 
Thus, having u ^= y"^ — gx = o, 

(du/dy) = 2j, {du/dx) = — 9, 
and at the point (4, 6) 

(du/dy'') = 12, (du/dx^) = — 9. 
Therefore, we have 

i2(y — 6) = g{x — 4) for the tangent, 
and -- 9(^ — 6) =12(^^ — 4) for the normal. 

EXAMPLES. 

Find the equations of the tangent and normal at the point (x', y) 
on each of the following curves: 

jj/ — J)/' = — (Px'/a^y'){x — x'), 

y — y ^ {a^y /b'^x){x — x). 



I. ^y + b'^x'^ = a^\ 



3. ^2+y = i?2. 



5. y = a\og sec (x/a). 



1 

\ y -y — -{y' h 

\ 



y-y = -{y ' lP){x - x'). 

y'y -f- x'x = R"^, 

y = W /x')X' 

4. y"^ = 2^*?^ — ;«r«. y — y z=: [R — x'){x — x')/y . 
y — y =z — cot {x /a){x — x'), 
y —y' =z tan (xycr)(jr — x'). 

6. ;rV3 +J1/2/3 = ^2/3. J r= - ^ ^jV^ + f ^"^V • 

{V = Qx/2 — '^/2, 
>' = - 2^9 + 29/9. 

5. X == r vers- ^ \/2ry—y^. y—y' = — (jf — y). 



1 



TANGENTS AND NORMALS. 245 

9. y^ = -^^. y-y=± ^^(3^ - ^'\ ^ - ^'), 

^/ 2a- X ■" ■" (2a - y)3/2 



10. tan -1- = k log |/x2 -j- y"^. y — y' = —, f^,(x — x'). 



\\. y — ae=^l<=. y — y — [y' /c){x — x'). 

12. a;f^ + 2bxy -\- cy'^ + 2i/x -|- 2^?)/ -}- /= o. 

ax'-\-h' + ^ , 

y—y= — T-r-\ r-r-{x — x), 

bx Ar cy -\- e 

13. ey — x. y — y —ip^ — x')[x' . 

14. e"^ = sin X. y ~ y' — cot x(x — x'). 

IC). xy = a. y —y' = — (y /x')(x — x'). 

16. y = 2/^;^ + r^'X^' y ~y' = — r-=-(x — ;f').- 

\2px' -\- r^x"^ 

8^3 ( ^-1-2;/ = 4a. 

4a^+j;^ \ y - 2x = - 'ia. 

ax'' \y= ±3 1/3-^/8 - a/8. 

'^^ + «- ( ^ = T 8x/3 ^3 + 4i«/36. 

ig. y{x—i){x — 2) = x —3. {x' = 3±V2) 

y= 1^2/(4 + 31/2), x = o/o. 

20. 3/ + j;2 = 5. (jc' = l) jJ' = T .29X ± 1.44. 

2 ^ (J^ = [3(^ + ^0 -5]//. 

21. jj/ == 6jr — 5. -j 

(jj/= -y'ix-x')/3-\-y. 
y=± (^+i)/2. 



22. y'^ = 2x'^ — x^. (x' = I 

' (y = T 2x ± 3. 

23. jj/ = (W^ + e-^/^y/2. V — / = {(="'!' — <f- V^)(jc — jtr')/2. 

24. X = ^ log [(a +4/;^^:r/)/y] -4/^^ -ji'2. 

y-y = - W/{^' -y0V2](^ - ^'). 

25. Find the angle at which x"^ =jj/^ -[- 5 intersects 8-r^ -{- i8jj/^ = I44„ 

Ans. 7C/2. 

26. Find the equation of the tangent to the curve x'^{x ~[- y) = 
a'^{x — y) at the origin. Ans. y = x. 



246 DIFFERENTIAL CALCULUS. 

27. Find the angle of intersection made by the two curves 

x^ -\- y^ — {laf, and j'^ — x^ — — a^ . Ans. tan-i|/i5. 

28. Find the angle at which the curve {y"^ -|- x"^)"^ = 2a'^{x^ — y^) 
cuts X. Ans. 7t/2. 

29. Find that point on an ellipse at which the angle between the 
normal and diameter is a maximum. Ans. {a/\/2, b/ ^2). 

30. Find the point on a parabola where the angle between a straight 
line to the vertex and the curve is a maximum. Ans. x = p. 

To find the equation of the tangent to y =/(^) which makes 
a given angle, 0, with X: 

Let a — tan 0; then dy' /dx' = a, which combined with 
y = /(x) w^ill determine {x\ y') for the point of tangency. 

Thus, to find the equation of the tangent to y = 6x 
which makes an angle of 45° with X: 

dy'/dx' = s/y = I ; .'• 7' = 3- 
Substituting in^ = 6x, we have x' = 3/2, and the required 
equation is j — 3 = x — 3/2. 

EXAMPLES. 

1. Find the equation of the tangent to y = a -\- (c — xy which is 
parallel to X. Ans. y = a, x = 0/0. 

2. Find the equation of the tangent to x^ -\-y^ = f^ which is 
parallel to y =: 2x -{- 7. Ans. y ± r/4/5 = 2{x T 2^/1/5). 

3. Find the equation of the normal to y'^/4 + ^Vq — ^ which is 
parallel to 2x — y = 3. Ans. y = 2x T 2. 

4. Find the equation of a tangent to iSy"^ -\- Sx"^ = 72 which is 
perpendicular to the right line pafsinof through the positive ends of 
the axes. Ans. 2y = 2^ ^ 1^97- 

5. Fmd the equations of the tangents to gy"^ — 2'^x'^ = — 225 
which makes an angle of 60° with the transverse a.^is. 

6. Find the points on y = x^ — 2^^ — ^^x -\- 85 where the tan- 
gents are parallel to X. Ans. (4, 5) and (— 2, 113). 

7. Find the equation of the perpendicular through the focus of a 
parabola to a tangent at {x' , y'). Ans. y = — y'{x — //2)//. 



TANGENTS AND NORMALS. 



247 



8. y^ = 4ax + 4^^i find the locus of the points of intersection of 
tangents, and perpendiculars upon them from the focus. 

Ans. X = — a; y = 0/0. 
g. y — 2 = (x — i)^x — 2; find the angle at which the curve 
cuts X, and the point where the tangent is perpendicular to X. 

Ans. Tan-i 2; (2, 2). 
10. Show that normal to y^ = 4ax is tangent to y'^= 4{x — 2afJ{2']a), 

With oblique axes, inclined at an angle y5, dy' /dx' repre- 
sents the ratio of the sines of the angles which the tangent 
and curve at {x\ y') make with the axes. The equation of 
the tangent remains unchanged in form, while that of the 
normal becomes 



y —y =. 



(,+,os/5|;)(.-y)]/[cos/j+g]; 

149. Let TM, PM, and NM, respectively, be the tan- 
gent, ordinate, and nor- 
mal at any point M 
whose coordinates are 
W.y'Y 

be perpendiculars from j 
the origin to the tan- 
gent and normal respectively. 

Let XTM = (f). Then, since dy' /dx' — tan 0, we have 
Subtangent = PT = / cot =/dx'/d/. 
Subnormal = PN — y' tan =-y'dy' /dx' . 

Normal = NM — y' sec cfy — y' \^i -\- tan' cp 




=/ Vi + {d//dx'y 

Tangent = TM = / cosec = / Vi + cot' 



= y Vi + {dx'/dyy. 



* Todhunter's Calculus, page 2S3; C Smith's Conies, § 42= 



248 DIFFERENTIAL CALCULUS, 

Also, from figure, or equations (i) and (2) (§ 148), 
OT =^ Intercept of tangent on X = x' — y'dx^ Jdy', 
OS = Intercept of tangent on Y = y' — x'dy' /dx\ 
ON = Intercept of normal on X = ^' •\- y' dy' / dx' , 
Hence, 

p — OQ = Perpendicular to tangent = OS cos 

_ ^n I _ y — x'dy' /dx' __ y' dx' — x' dv[ 

~ V 1 4-tan=^ (f)~ Vi^ {d//d^~ Vdx" -h dy^'' 
q = OR = Perpendicular to normal = ON cos 
_ x' -\-y'dy'/dx' __ x'dx' ^y'dy' 
~ \\^{dy'/dx'y ~ Vdx'-' + ^/^* 
To apply these formulas, obtain general expression for 
dy' Jdx' from the equation of the curve, and substitute 
values of x* and f. 

EXAMPLES. 

1. //^^ + x'Ja' =1. / = aHyib'x"' + «y 2)1/2. 
Subt = - ayy{Px') = {x"" - a})lx. Subn = - Px'/d'. 
If a ^= b, we have for the circle r ^ a, p =: a, Subt = — y"'/x' , 

Subn = — x' . 

2. y-'/b"" - x'^Jd' = - I. / = - aUyib^x"' + «*y'2)V2. 
Subt = ay V(^*x') = {x''^ + a'V^'. Subn = b'^x'/d'. 

\ia =zb, we have for an equilateral hyperbola ;> = d^/i/x"^ +y^ 
Subt = y'^/x'y Subn = x'. 
Z. y = 2px + r^x'. 

Subt = {2px' + rV2)/(/ 4- rV); Subn = / + ^-V, 



Ta„ = /... + .V=H-(^^fl±Z.-)'. 



Nor = \ipx' + rV^ + (/ + r'x'Y 



If r2 = o, we have, for the parabola referred to axis and tangent at 
vertex, 



Subt = 2x', Subn = p, Tan = y'^/p'' -^ y"^/p, 



TANGENTS AND NORMALS. 



249 



Nor = yy''+/^ Perp. to tan=>/'2/24/y'^-|-/^ q=y\x'^-p)l ^^^p. 

4. xy = m. 

Subt = — y, ■ Subn = — yyy= ~ y^/m, 

Nor =y|/x'2 -f yy^', Tan = V/M^". 

5. j^/ = fz*. Subt = i/log a = i^a. 
5. X = r vtxs-'^iy/r) — ^2ry — y , is the eq. of a cycloid.* 



Subn = ^ iry' — y'^ 



Subt = y-'l^/iry' - y'\ 

Tan ■= y ^ 2ry' I ^ iry' — y'^, Noi = j/ary' 
Y 




P N A X 

Since the subnormal PN = MD, the normal at any point passes 
through the foot of the vertical diameter of the corresponding posi- 
tion of the generating circle, and the tangent passes through the 
other extremity. 

The tangent and normal at any point of a cycloid are therefore 
readily constructed when the corresponding position of the generating 
circle is drawn. 

Otherwise, when the circle AB, upon the greatest ordinate as a 
diameter, is drawn, through the given point i^draw MC parallel to 
the base 0X\ from C where it cuts the circle draw the chords CB 
and CA to the ends of its vertical diameter; through M draw the 
tangent ME parallel to CB, and the normal J/TV parallel to CA. 

To construct a tangent parallel to any right line as RS, draw the 
rhord BC parallel to it, through C draw CM parallel to OX, and 
through M draw ME parallel to RS. 

7. y"^ = a'^-'^x. Subn —y"^/nx', Subt = nx' . 

'■"' Prith described by a point on circum, of a circle rolling upon 

a fixed rii/ht line. 



250 DIFFERENTIAL CALCULUS, 

8. f' = ax" + x\ 

Intercept of tan on y = \x' [{a + ^')f '^(^/s)- 
^9. ;>/' = IX. y = 8, Tan = 4 |/i7. 

10. X = sec 2jf/./ = [{x^ - 1)1/2 sec-i^f - i]/[i + 4;t:2(;tr« - i)^^^]. 

11. y^ = x^/(2a — x). 

Subt = x'{2a — x')/{2a — x'), Subn = ^(sa — x')/{2a — x')^o 

Nor =/'Vc-, Subn = c{e^=^'/c) - e-^^'/c)/^. 



Tan = j"V l/j'* — tf", Subt = cy'/\^y'^ — c^. 



13. ^* +JJ/* = <r*. / = c^/^x'^ + /«. 

14. ;c^ — "^axy -\-y^ = o. 

£q. of tan jj/ = — ;»;(x'2 — «;'')/(/' — a^') + («^y)/(y' - ax'\ 
Subt = (_j/'3 — axy')/{ay' — x'^). 

Eq. of tan y = -x ^{y'/x) + f ^. 

Intercept on X = |/aV, on F = f^a^^ . 
Length of tan between axes = a. 



Area between axes and tan = \/a^x'y' /2. 

p = \ ax'y' . 
\^ y - ce^f''. Subt = a, Subn = f^/a, 

17, y = 2^^^ log x. Subn = «Vj;'. 

18. V = a2(a - ;c). Subt = — 2{ax' - x"^)/a. 
ig. 3^!^ -|- a-'' = 2x^. Subn = x'Y^. 

x' = 4, Subt 

Subt = 2x'/2y Subn = 2>(^x"^/2. 



TANGENTS AND NORMALS, 2$ I 

POLAR COORDINATES. 

150. Polar Tangent, Subtangent, Normal, and Sub- 
normal. 

Let PM =. r, corresponding to th'fe point of tangency M 
of any tangent as TM. From the pole P draw PT per- 




pendicular to PM. Draw PQ and MR perpendicular to 
TM, 

PT^ the part of the perpendicular to r, from P to its 
intersection with the corresponding tangent 7W, is the 
polar subtangent corresponding to M, PR is the polar 
subnormal, and RM is the normal. 

Let = PMT, the angle made by r with TM, 

Then (§ 55, § 70) 

From Trigonometry, 
sin = tan 0/ Vi -p tan^ 0, cos = Vi/(i + tan^ 0). 



Therefore, since tVr' + rV<9^ = ^^, (§ 92,) 
sin = rdO/ds, cos = dr/ds, cot = dr/rdQ. 



252 DIFFERENTIAL CALCULUS, 

Hence, 

PT=. Subt = r tan (p = r'dO/dr, 



TM = Tan - ;Vi + {rdid/drY = rds/dr, 
PR = Subn = r cot = ^r/^6', 



PM = Nor = l/r' + {dr/ddf = ^V^iy, 
/'(2 =/ = perpendicular to tangent 

= r sin = rdS/ds 



V(dr/dOy -f r" 

Since ^j/^9 is assumed to be positive, positive values only of/ are 
considered. 

i// = i/r' + (dr/dey/r\ 
PuttiT^g j/r = Uy from which dr"^ = r^dji^ we liave 

i// = ^^^ + (^2//^6')' (i) 

PN =^ q = perpendicular to normal = r cos 



= rdr/ds = =r = V/^ — /. 

j/i + {rdO/drf 



Since </^ =^ dr/{dr/ds) = rdr/r cos = rdr/'q^ 
ds/dr = r/q = r/ S/r" -/; /, ^j = rdr/ \f7 
But / = rV^M, .*. ^i- = r'dB/p. 

Therefore r'dd — prdrl ^ r" — p\ 



TANGENTS AND NORMALS. 253 

EXAMPLES. 

1. r = a0; find the angle between r and the tangent, the sub- 
tangent, the subnormal, the normal, and/^. 

(p = tan-19, Subt = r'^/a, Subn = a, 
Nor =Ya2 + r\ p - r""/ ^d" + t"". 

2, r = a{i -j- cos 0). 



Subt = — rV(a sin 0), Subn = — a sin 6, / = \fry2a. 

3. r = a^. Subt = Mar, Subn = r/il/a, Tan (p = Ma. 
Tan = r ^T+M^\ Nor = r |/i + i/i^ / = i^a ^ l^i^a' + i • 

4. r'^-a'J^. Subt =2a 4/0^/=2aV/i/r* + 4a* =2a |/e/|/i + 46^. 



5. r = fl0i/2. Subt = 2rV^^ / = 2rVV«* -\-\r^ 

6. ;' = 2(3! cos 0. 

Subt = — 2a cot cos 0, Subn = - 2a sin 0, 

Tan = 2a cot 0, Nor = 2a. 

*]. r — fl0-i. Subt = — a. Tan = — 0, Subn = — ry^. 

8. r = a sin 0. = 0. 
g. ?''* = a"^ cos 20. 



= It/ 2 + 20, Subt = — r^l^a} sin 20), Tan = rd^/ i/a* — r*, 

Subn = — a2 sin 20/r, Nor = a'^/r, 

p — r^/a"^, Perp. to Nor = r sin 20. 

10. r = «0/sin 0. Subt = rt0V(sin - cos 0) 

11. r = ab/{ae^ + be-^). Subt = - abjiae^ - be'^). 

12. r — a{\ — cos 0). 

(p = B/2. Subt - 2a sin^ (0/2) tan (0/2), / = 2a sin^ (0/2). 

13. r = ae^ cot a, /* = r sin or. 

a(l - e^) a\l - e^) a''{l - (''Y 

I — (f cos ^ 2a/r—i I — 2<f cos + <?' 



254 



D IFFEREN TlA L CA LCUL US. 



15. r = a -B- 1/2. 
«(^2 _ i) 



16. r = 



p = ia''r/»i/r^ + ^a\ 
b i/r 



1 -\- e cos 0* 



151- 




y2a -\- r 

Polar Equation of a Tangent.— Let PM = r\ and 
XPM= e\ be the coordi- 
nates of M, the point of 
tangency of any tangent, as 
NM, and let FN = r, and 
XFN = e, be the coordi- 
nates of any other point of 
NM,2.^N. l.ttPMN=c()', whence (§150) tan 0=rV^7^r'. 
Triangle FMN gives 

r^ _ sin MJVF _ sin [{6' - 6^) + 0] 
/- ~ sin FMN ~ sin 

= sin (^' - ^) cot + cos (/9' - 6) 

= sin (<y' - 6) dr'/r'dd' + cos {B' - 6), . 

Putting i/r = u and i/r' = u\ whence 
du'/dS' = - dr'lr"dd\ 

we have, dividing both members of (i) by r\ 

u = u' cos (6^' - 6*) - sin (^' - 6') ^z^'/^^' 

for the tangent. 

Otherwise it may be obtained from eq. (i) (§ 148) by 
changing the reference to a system of polar coordinates. 

Let r' and 0' be the polar coordinates of the point of 
tangency {x',y'), and r and 6 those of all other points of 
the tangent. Taking the pole at the origin and the refer- 
ence line to coincide with the axis of X^ we have, as in 
ex. II, § 115, 



(0 



(2) 



TANGENTS AND NOEMALS. 2^% 

X = r cos d, x' = r' cos 6\ 

^ = r sin 6^, y' 

d£_ _ sin B'dr' + r' cos B' dB' 
dx* ~ cos e'dr' - r' sin ^V^'' 

Substituting in (i) (§ 148), we may write 

. . , . ^, sm 6' dr'/de'-\-r' cos 0\ ^ , ^,. 

r sm d—r' sin 6"= ^, , , , ,^, ^-T--27>(r cos 6/—^ cos t) ), 

cos o dr /dd' — r sm c/' 

Whence 

r^, sin cos 9' - r'^, sin 0' cos 0' - rr' sin sin 0' + r'^ sin^ 0' 
do ao 

= r^, sin 0' cos + r/ cos cos 0' - r'^ sin 0' cos 0' - r'^ cos^ 0', 
or 

——(sin cos 0' — sin 0' cos 0) — r/(sin sin 0' + cos cos 0') = — r'', 
ao 

or, changing signs of terms, 

rdr' 

— ^ (sin 0' cos - sin cos 0') + rr'(cos 0' cos + sin 0' sin 9) = /«, 

or sin {B' -B)dr' /rUB' + cos (^' - ^) = ^'A- . (i) 

Putting i/r = ^/, and i/r' = u\ whence, 

du'/dB' = - dr'/{r"dB'), 

we have, dividing both members of (i) by r', 

u = u' cos (^' -B)- sin (^' - B)du'ldB\ 

Representing the polar coordinates of all points of a nor- 
mal except the point of tangency (x\ y') by r and B, we 
deduce in a similar manner from (2) (§ 148) 

u^u' cos (^ - 6") - 2^'^ sin (^ - B')dB'/du' 

for the normal. 



256 



D IFFEKEN TIA L CALCUL US. 



CHAPTER XIII. 

ASYMPTOTES. 
RECTANGULAR COORDINATES. 

152. An Asymptote to a curve is a definite limiting posi- 
tion of a tangent to the curve, under the law that the point 
of tangency recedes from the origin without limit. 

Unless otherwise mentioned rectilinear asymptotes only 
will be considered. 

Asymptotes Parallel to the Coordinate Axes. 
153* If in the equation of a curve >/:^->oo as x'm-^a, then 
(dy/dx)a =^ (§ 71), and x^a is the equation of an asymp- 
tote parallel to Y. If, otherwise, ym^a as ^b-^oo, then 
^ = <z is an asymptote parallel to X, 

EXAMPLES. 




\. y^ — x^/{2a — x). 
ym-^ ± 00 as x'!i»-^2a. 
Her4ce x = 2a is an asymptote. 



ASYMPTOTE"^ 257 

2. <fl/* =_j/— 1. ;j/B->I as ^^-^±00. 

Hence ;j/= i is an asymptote to both branches. 

_)/:^->Go as — x'^-^Q. 
Hence F is an asymptote to the left-hand branch. 




3. jjr = a*. ym-^o as xb->— 00 

Hence, Xis an asymptote. 



4. ^ = 



23^.;*; + Pc 






«7x2 - 1 • 

jI'B-»oo as xB-^lrt', and ji'S^-»o as jcB-^oo, 
Hence, X and ;r = ± a are asymptotes. 

CURVES. ASYMPTOTES. 

5. j/2 == {x^ -|- ax^)/{x —a). X = a. 

6. xj/ — ay — bx ■=■ o. x ■= a, y = b. 

7. «V ~ ^"'y ~ ^^^- X = ± a, y = o. 
8 y = x/{iJr^'). y=^o. 

9. X = a log [(rt; + Va^ -f)/y] - V^^^- V = O. 

10. y— a^x/{x — of. X — a, 7 = 0. 

11. ;/ = log X. X = O. 

12. x^ -f- a^^j* = axy. x = b. 
^'x. y ■= a'^x !{a^ -\~ x"^^. y = o. 
14. xy- = 4rt;-(2a — x'^). x = o. 



258 



D IFFEREN TIA L CA L CUL US. 



15. 
16. 

17. 

18. 
19. 
20. 
21. 
22. 

23- 
24. 



x^y -\- a^y = a^ 



25. y 



x=±{y + b)\/a''-yyy. 
y = a^/{a^ — x"^). 

y(a^ -\- x'^) = a^(a — x), 
y = e-^. 

y = a-\- b^/{x - c)K 
a?y -(- x^'^y = a^x. 
x^ — 3^ = 6xy. 
y(a^ - x"") = b{2x + c). 
2 _ a^jx — d){x — 3<^) 



x"^ — 2a;^r 



^ = 0. 

jc = ± «, ^ =0. 

>/= ± «. 

J>' = 0. 

j(/ = o. 

y •= a, X r=zc, 

J' = o. 

:r = o. 

_j/ = o, X — ±a. 

^ = 20!, JP = o, ^ = J, a. 



ASYMPTOTES OBLIQUE TO THE COORDINATE AXES. 

154. Let JJ/be a tangent and SP an asymptote to any 
plane curve, as NM. 




The equation of TM'x?, (g 148) 

and its intercepts are 

OR^x' - y\dx'ldy'), OT = y' - x'{dy'/dx'). 



ASYMPTOTES. 259 

As the point of tangency M recedes from the origin 
without limit, the tangent TM approaches the asymptote 
SP, and the intercepts OR and OT approach OS and OP^ 
respectively, as limits. 

Assuming that in the equation of the curve y :^-^ co as 
x' ^-> 00 , and placing 

tan OSP = K, OP = Y,, and OS=X,, 

we have, omitting the dashes, 

limit p^-j limit p ^-| 

limit 



^ ^""'^ r dx 

. .^^^ ^ L ay _ 



y m-^ CO 

any two of which will serve to determine in regard to an 
asymptote. 

If ^ = o or 00 , or if F„ = Xo = 00 , or if either Y^ or 
X^ is imaginary, there is no corresponding real asymptote 
within a finite distance from the origin ; otherwise there is 
and its equation is 

y = Kx-\- F„, or x=y/K-^X,, 

or x/X,-^y/Y, = i. 

If either Y^ or X^ is zero, the corresponding asymptote 
passes through the origin and its equation is 

y = Kx. 

EXAMPLES. 

I. y = 2J}X. 

X = CO = y, dy/dx = p/y. K ■=■ O. 



26o 



D IFFEREN TIA L CALCUL US. 



Hence, a parabola has no asymptote. 



X\ 



± <x> , jj/ = q: CO . dy/dx = {4a X — Sx^)/^)'^. 



2{2ax'^ 



4ax — 3^ 

Hence, y = — x -{- 20/2 is an asymptote. 
2. y^ = 10 — x^. 

X = ± CO , y = Tco. dy/dx = — x^/yi, 

x. = [.+yA']^=[.oA']^=o. 

A-= [- ^vy]„= - [do -y)/y]r= -['°^' - ■]r= - '• 

Hence, jj/ = — ^ is an asymptote. 

jr = 00 = JJ/. dy/dx = (^ -|- rs^ir)/ \2px -f rV2. 



JTo = 






/A + 



-1 ==^- 



When r'^ < o, Fo is imaginary. Hence, an ellipse has no asymp- 
tote. 

When r^ = o, both results are unlimited, showing that a parabola 
has no asymptote. 

When r"^ > o, both results are finite, and since Fo has two values. 
an hyperbola has two asymptotes 

Putting/ = b'^/a and r^ = b'^/d^, we have 

J^o = - «. Fo = ± b. 



X, 



_ 3V- -] 

2^/jr -)- 3-^'M ^ 



'za -f- 3x 



dy/dx — {2ax -\- 3x2)/3_j/2. 

>lx 4- 3) I ~ ~ T 



(2,. A + 3) 



ASYMPTOTES. 

2ax^ + 3JC^" 



261 






x'-ff' 



Hence, v = ^ + «/3 is an asymptote 

6. ^3 _ ^ = ;,o _ 4;,. ^jf/^^ = (3^^ - 4)7(3/ - I). 

Put X = 0^, giving / - >' = ^y — 4<y. 

Hence, ^ = ± ^{M^^mf^^^ and / = i give;.= ± «. = ^. 



Xo = [--7(|^)]^^- 



3^^ 



[-3/(3^-4A)]±- =°- 



ir = 



3^- 



V 



- I J±co L3 - I A' J ±00 



Hence, j>/ = ;c is an asymptote. 



7. ^' 



3^;,^ +/ = o. ^^A-^ = («^ - ^^'VC/ - ^^)- 



Put .^f = ^J, divide by /, and we find y = 3^//(i + ^^), in which 
^ _ _ I gives ^ = CO = — X. 



Xn= X 






r y - axyr\ _ r - ^^n = r_z.^ 

_ r - ^ "1 ^ r _ j^rf^i^n ^ K 

- - Li- alx \~ L / - ^^ Jco 



Hence, jj/ = — ^ — a is an asymptote. 

ASYMPTOTES. 

_j/ = ± bxja. 
y — — X -{- a/3. 



CURVES. 



9. y = x2(a — x). 

10. y = ax + ^•^t^''*' 

11. / = xy{x - I). 

12. y^ -{- x^ = 3-^'- 

13. y = ;cV(x2 + 3«'). 

14. y = a^ — x3. 



_j/ =^ \/bx + a/2^b. 
X ■= 1. 

y = — X -\- 1. 
y = X. 
V = - X. 



262 DIFFERENTIAL CALCULUS. 

15. y^ = a^x — x^. y ■=. — x. 

16. y'^ — 6x^ -{- x^. y = X -\- 2. 

17. xy = x^ -\- X -{-y. X = — I, ±y = x. 

18. Find the perpendicular distance from the focus of an hyperbola 
to an asymptote. Ans. Semi-conjugate axis. 

ig. Find a tangent to a given curve which forms with the coordi- 
nate axes a maximum or a minimum triangle. 
Let ul2 = area of triangle. Then 

dx'\l , dy'\ I , dv' y dx' 



'TyK ~"^;="r "" dx' ] df 



-{^■-^fm^^)zim 



du 
dx 

dy , , ,dy 

- ,.) - = O and y — x —- 

dx '■ dx 



y du . ,dy , ,dy , 

general, - — = o requires x -—7- -\- y = o, whence y — x ^, — 2y . 
dx dx dx 

Therefore, in general, when the triangle is a maximum or a min- 
imum, the portion of the tangent between the axes is bisected at the 
point of tangency. 

Let the equation of the curve be x^ -{- y^ ■= R^. 

Then C = :i^, and -''*'--''' 



dx y dx' y' 

Hence, — -j— +/ = o gives / = y, 

and u/2 = A'^ is a minimum. 

155. The equation of any algebraic curve of the nih. 
dec^ree may be arranged in sets of homogeneous terms and 
v^ritten in the form 

xVXy/^) + ^'^-y.(jA) 4- x--'f,{y/x) + . . . = o. (i) 

Thus, having 

y - x'y + 2/ + 4_)^ + ^ = o, 



ASYMPTOTES. 263 

then (/ - x'y) + 2/ + (^y + ^) = o, 

and x\flx' - y/x) + x\2flx') + x^^y/x + i) - o. 

Let y — mx -\- c\>Q the equation of a right line ; combine 
it with (i) by substituting mx + c for j, giving 

-f :^«-y>^ + ^A) + ... = o, . (2) 

the n roots of which are the abscissas of the n points com- 
mon to the two lines. 

By causing in and c to vary, the right line may be made 
to have any position in the plane XY. 

Developing each term of (2) by Taylor's formula, we 
have 
x^fXm) + x^-\cf:(in) +/,(;;0] 

+ ^'^-f 7/o"W + cf:{jn) +/,(/;0] + etc. = o. (3) 

Any set of values of m and c which satisfy the two equa- 
tions 

fXni) = o, .... (4) </;w +/,(«) = o, (5) 

cause the two terms in (3) containing the highest powers of 
X to disappear, and the resulting equation to have two in- 
finite roots. That is, two of the points common to the 
right line and curve are thus made to coincide at an infinite 
distance from the origin, and the corresponding right line 
consequently is an asymptote. 

Let 7n^^ m^^ ... nin be the n roots of (4); the cor- 
responding values of c from (5) will be ~/i(^i)//o'(^i), 
etc., and 

y =z m,x -AM///M> ' 



264 DIFFERENTIAL CALCULUS, 

y = m^x -f,{m^)/f,'{m^), 
etc. etc. 

y = MnX — /i(w«)//o'(^«), 

are the equations of the n asymptotes. 

Hence we have the following rule : 

In the equation of the curve substitute mx -\- c for y, place 
the coefficieitts of the two highest powers of x equal to zero^ 
and find corresponding sets of values for m and c. Each set 
will determine an asymptote real or imaginary. 

Thus, having x^ — 2xy — 2x^ — Sy — o, then 

x^ — 2x'^(mx -{- c) — 2x'^ — S{mx -\- c) = 0, 
and (i — 2m)x^ — (2^ -f~ 2)^^ — Zmx — 8^ = o. 

I — 2m. = 0, — 2^ — 2 = o, give m =1/2, ^= — 1. 
Hence, y = x/2 — i is an asymptote. 
Having xy"^ — x^ — ay^ — ax^ = o, then 

x{mx -{- cY — x^ — a{mx -\- cY — ax^ = o, 
and {m"^ — i)^^ -1- [2mc — am^ — a)x'^ -\- etc. = o. 
m^ — 1=0, 2mc — am^ — ^ = o, give ;;z = ± i, <: = ± ^. 
Hence, >» = ± (^ -|- ^) are asymptotes. 

EXAMPLES. 
CURVES. ASYMPTOTES. 

1. y\x — 2d) = x^ — a^. x = 2a, y = ± (x -{- a). 

2. y = x(x -f- i)V(x — i)^ X = I, y = X -j- 4. 

3. X* — y* = a'^xy. y =. X, y ^ — x. 

4. X* — x'^y^ -}- a'^x'^ -\- i>* = o. x=—y, x=:y, x = o. 

5. x^ ~\~y^ -\- 2)d^x + 'ib'^y A^ a = O. X = — y. 

6. x^ -\- y^ = sax^y^. ■^ -\- y = a- 

7. xy^ — ay* -f" •^!y^ = P- x = a, y = — x — a, y = o. 



A S YMP TOTES. 265 

x^ =iO. y ^= — X, y -\- \ =. X, X =■ o. 
9« (/'^ — 7>^y + '2-x'^)x — 4ay^' x = 4a, y = x — ^^a, y = 2{x -f- 8a). 
10 x(y -\- xfiy — 2x) = a*. x = o, y = 2x, y = — x. 

11. / = x''{x^ - i)/(x2 4-1). y = ±x. 

12. y^ — x'^ = a^x. y = X. 

13. y^ = {x^ - a'x'')l{2x -a). x= a/ 2, y = {x -\- a ft)/ )/2. 

14. x^ — xy"^ -\- ay"^ —O. x = a, y — ± {x -\- a/2). 

15. y"^ = x\x^ — 4a^)/{x'^ — a"^). y = ± x, x = ± a. 

16. y = x{x — 2a)/(x — a). y = x — a, x ^a. 

17. x^ -\- 2xy — xy"^ — 2y^ -(- 4y -|- 2xy -\- y = I. 

X -^ 2y = O, X -\-y = 1, ^ — ' = — I. 

18. y^ —6xy^ -|- iix^y —6x^ -\- x -\-y = 0, y =z x, y ^ 2x, y = ^x. 

19. y^ — x'^y -\- 2y'^ -{- 4y -\- x = o. y=o, y = x—i, y=— x—i. 

20. y* — X* -\~ 2ax'^y — b'^x := O. _y = x — a /2, ^ = — x — a/2. 
21 X* — y* — a^xy — Py"^ = o, x = ^ y. 

22. x'^y Ar y'^x — c^, x = o, y — o, x = — y. 

23. 2x^ — x'^y — 2xy'^ -\-y^ -f- 2x^ -\-xy— y'^-\-x-\-y-\-i =0. 

_y = x-|-i, _y = — X, y =[2x. 

156. Let the equation of the curve (i) (§ 155) be arranged 
according to the descending powers of x; thus: 

ax^ + {^y 4- d)x''-'^ + . . . = o. . . . (6) 

If a = o andjvbe assumed equal to —d/b^ two of the roots 
of (6) are infinite and the right lineji' == —d/b is an asymptote. 
Hence, when x^ is missing, the coefficient of the next highest 
power placed equal to zero is the equation of an asymptote 
parallel to X. 

If both x^ and x^~^ are missing, the coefficient of .x"~^, 
which will be of the second degree with respect to j', placed 



266 



DIFFERENTIAL CALCULUS. 



equal to zero, determines two asymptotes real or imaginary 
parallel to X. 

In a similar manner asymptotes parallel to Y may be 
determined. Thus, having x"'/ ^x^y—xy^ -\- oc-\-y -\-\ =o, 
in which x^^ x^^ y\ y' are missing, y^ —y is the coefficient 
of x^ and x^ — x that of y^. 

Hence, y — ^ = o, x^ — x = o, give the asymptotes 
y = o, _y = I, X = o, and x = i. 

EXAMPLES. 
CURVES. ASYMPTOTES. 

1. y'x — ay— x^— ax'*' — ^^ = o, ;r = a, y ^= x-\-at y =— jc— <», 

2. xy^ + -^^y — a* = o. j^ = o, :f = o. 

3. ^y - a^C^t'^ -\-y'') = 0. :«r = ± d!, J = ± <j!. 

4. j^y — a2j2 — ;i; = o. :v = ± a, _^ = O. 

5. ajj/2 — ^rj/'* — ;t:^ = o. ■ X = a. 

Polar Coordinates. 
157. If for any finite value of ^, designated by a, /^ is infi- 
nite, and the corresponding subtangent -P7"=(^V6^/^r)fl=«, 




designated by -S", is finite, the curve has an asymptote paral- 
lel to the radius vector corresponding to a. 



ASYMPTOTES, 



267 



Knowing a and ^S", the asymptote may be constructed as 
follows : 

Through P draw PM\ making the angle XPM' = a; it 
will be the direction of the corresponding infinite radius 
vector. Draw PT' perpendicular to PM', and lay off 

pr= S= (r'dd/dr)e^a, 

T'M' drawn parallel to PM* is the corresponding asymp- 
tote. 

Designate the angle XPT' •= a — 71/2 by ^, and let r 
and d be the polar coordinates of all points of the asymp- 
tote. Then from the triangle T'PR we have 

r = S/sin {oL-O) (i) 

Hence, knowing a and 6*, the polar equation of the 
asymptote is known. 

In some cases a and S may be readily found. Thus, having 
the curve r = ad /sin 6^ 6 = tt = a gives r = co ^ and 

ad' \ ^ ^^ 

;sin 6—6 cos 6^/^=77 

Hence, r sin 6^ = <27r is the polar equation of the asymptote. 
Positive values of S are laid 
off in the direction 

(3= a— 7r/2, 

and negative values in the 
direction 

/?+ i8o° = a-}-;r/2. 

Thus, having r = a/0^ 
6= o = a gives r = 00 , and S = — a. Hence, P =— ^/2, 
and direction of S is 71/2. r sin 6 := a \s the equation of 
the asymptote. 



\ dr )0=a \s 




268 



DIFFEREN TIA L CALCUL US. 
M 




Flavin g r = a -\- a/Oy 
6 =^ o =z a gives r = oo , and 

S = {r'd6/c/r)e^o ^ - a. 
Plence, r = «/sin d is an 
X asymptote. 

As 6*^-^00, r-m-^a, and the 
circle R = a \s called an 
asymptotic ci7'cle. 
iSS* In general, let the polar equation of a curve be 

r"/„(&) + r--'fn-.{fi) + . . . + r/.(&) +/,(^) = O. (l) 
Placing u = r/r, clearing of fractions, and arranging with 
respect to u^ we have 

«y.(^) + t^-vm + ... + ufn-m +/«(^) = o- (2) 

For any point corresponding to r = oo ?^ must be zero, 
and the roots of fn{P) = o are the values of Q correspond- 
ing to r = CO . 

Differentiating (2) with respect to B, making u = o and 
^ == Of, we have 

(du/de)^^^fn-x(pi) +/«'(«) = o. . . . (3) 
Therefore (§ 151), 

Hence (i) (§ 157), the polar equation of the asymptote is 
r sin {oL — d) = fn-i{oc) //n\ay . . (4) 

When « = I, 

rsm(a^e)=Ua)/f^\a) (5) 



A S YMP TO TES. 269 

To illustrate, having r cos ^ — ^ sin ^ = o. 

;2 zz. I, f^B) =: cos e, fXO) = -bsin e. 

cos 6^ = o gives a = 7t/2, 37r/2, etc. 

y^(<ar) = — <^ sin a, //{^) = ~ sin ex. 

Hence, r sin {7t/2 — 0) — — (d sin r-e)/ — sin or = ^, 

or r cos 6 ^ b ior the corresponding asymptote. 
Let r = a sec ^ + /^ tan ^. 

Here i/r = u = cos ^/(^ + ^ sin ^) = o 

gives a = ;r/2, (^^^/^^)^^^/^ = - i/{a + <5), 

and r cos ^ = ^ -f ^ for the corresponding asymptote. 

(4) maybe deduced from (2) (§ 151) by substituting in 
it u' = o, 6/' = a, and - du'/dd' ^ fn\oc)/f„.^{a). 

EXAMPLES. 
Find the asymptotes of the following curves : 

\, r = a tan 0. r cos 9 = ± <r, 

2. r cos = a cos 2O. r cos 6 = — <r. 

3. {r — a) sin = ^. r sin = ^. 

4. r cos 20 = a. r = tf/[2 sin (45* — 0)], 

5. r sin 40 = a sin 3O. 

6. r — a0-iA . r sin = o. 

7. /•' sin = 0' cos 20. y sin = o. 

8. r = a cosec (0/2») r sin = ± 2na. 

Q. rsin;z0 = «. nr sin (0 ) = --. 

\ « / cos /$;r 

(In which k is any integer.) 



2 JO DIFFERENTIAL CALCULUS. 

10. r = «(sec ± tail 0). r cos G = 2a. 

11. r =: 2a sin 6 tan 0. r cos 6 = 2a. 

12. r = cz sec niQ -\- b tan m^. r sin [7r/2w — 0] = (a -[- b)/m. 

13. r^ cos = rt^ sin 30. r cos 6 = 0, 

14. r = «0V(0^-i). r= -«/[2 sin(i8o77r- 0)]. 

15. r = rt;/cos + <5. r = «/sin (90° — 6). 

Find the asymptotic circles of the following curves : 

16. = — ■ circle r = 2a. 

\2ar — r''^ 

17. r = a0Y(02 _ i). circle r = «. 



DIRECTION OF CURVATURE. 



271 



CHAPTER XIV. 
DIRECTION OF CURVATURE. SINGULAR POINTS. 



DIRECTION OF CURVATURE. 

159. Let j^=/(^) be the equation of any plane curvej 
and let ^ represent the angle 
made by any tangent to the 
curve with X. 

Then dy / dx^ldoa 6*, 

and d'^y/dx' = (d tan 6)/dx. 

When d-y/dx^ is positive^ 
tan increases with x (§ 71), and, as may be seen from any 
figure, the concave side is above the curve and the direc- 
tion of curvature is upward.^ 




\l 


( 


A 




\ ^ 






/^ 


^0' 


v«^ ^ 





/ 


/ 


> 





When dy/dx^ is negative, tan 6^ is a decreasing function 
of X, and t/ie direction of curvature is downward. 

* The direction of curvature at a point is along the normal towards 
the concave side. 



2/2 



D IFFERENTIA L CAL CUL US. 



In a corresponding manner it may be shown that positive 
values of d'^x/dy' indicate concavity towards the right, and 
negative values concavity towards the left. 

To illustrate, let y ~ 2 -^ {x — 2)^ 

Then d-'y/dx'' = 6ix - 2), 

and d'^x/dy'' = 2/9(2 — x)^ 

Hence, when x < 2, the direction 

of curvature is downward and to 

the right. When x > 2, the curvature is upward and to 
the left. 

Show that the direction of curvature oi y = e'' is always 
upward and to the left. 

160. Polar System. — Representing by/ the perpendic- 
ular distance J^Q from the pole J^ to the tangent at any 





point M on a. curve, it is apparent (when the tangent does 
not pass through /') that the curve is concave towards 
the pole when p is an increasing function of r, and that it 
curves away from the pole when/ is a decreasing function 
of r. 

Hence, positive values of dp/dr indicate concavity towards 
the pole and negative values the reverse. 



DIRECTION OF CURVATURE. 



273 



From (i) (§ 150) we have i//^ = 21^ -j- {du/dOy, from 
which - dp/p' = (z^ + d'u/d6')du; but du = - dr/r\ 

Hence, dp/dr = {p^ /r'){u + d'^u/dO'^)^ which, since / is 
always positive, changes its sign only with {ii -\- d'^u/dO'^). 



EXAMPLES, 



I. Let r — aO *. 

Placing \/r = u^ we have u = ^/^, du/d6 = 6~-/2a, ami 

Hence, ^^ + d'u/dd' = {0^ - 6-^/4) /a. 




Therefore, when 6 < \, dp/dr is negative, and the concav- 
ity is away from the poie. 

When ^ = i, whence r =: a\/2^ dp/dr ■=^ o. 

When 6* > I, dp/dr is positive, and the curve is concave 
towards the pole. 

2. Show that r = p/{i — cos 6^) is always concave towards 
the pole. 

3. Show that the direction of curvature of r = a^ is 
always towards the pole. 



274 DIFFERENTIAL CALCULUS. 

SINGULAR POINTS. 

Points of any curve which, independent of coordinates, 
possess some unusual property are as a class called singular 
points. 

1 6 1. A Point of Inflexion is one at which, as the -variable 
increases^ a curve changes its concavity from one side of 
I the curve to the other. The cor- 

//^ responding tangent intersects the 

y\ curve, and the direction of cur- 

y/7 I vature is reversed. 

' |p Let^ = fx be the equation of 

a curve. It follows from § 159 
that c is the abscissa of a point of inflexion, if as ^increases 
f"x changes its sign in passing through /'V. 
The real roots of the equations 

f"x = o and /"x = 00 

are therefore critical values, and may be tested by a method 
similar to that described in § 135 in the case of fx = o 
and/'jc = 00. 

Hence, the general method for any critical value as c is to 
determine whether, as h vanishes from any definite value, 
f'\c — h) and f"{c-\- h) ultimately have and retain dif- 
ferent signs. 

When/"(x),/'"(jic:), etc., are continuous for values of x 
adjacent to critical values, those derived from f"{x) = c 
may be examined, as in the case of fx == o (§ 136), as 
follows : 

Having f'\c) = o, substitute c for x in f'"{x)^ f^^(x), etc., 
in order, until c result other than o is obtained. If the cor- 
responding derivative is of an odd order, c is the abscissa of a 
point of inflexion, otherwise not. 



SINGULAR POINTS. 



275 



If a result oo is obtained^ the general method should be em- 
ployed. 

When f'"x is complex, the general method is usually 
preferable. 

It should be observed that at a point of inflexion the 
slope of a curve is either a maximum or a minimum, and 
that the determination of such points is the same as finding 
those at which /"'(jt:) is a maximum or a minimum. 



EXAMPLES. 
\, x^ — ix^y — 2x^ =•■ Sy. 

dy _ x{x^ -\- \2x — i6) 

Tx~ i{x-' + 4)2 • 

(Py _ - \{x'^ - 6x^ - 12^ - 1 -8) 

dx' " (-^^ + 4j' 



o gives 



X=-2, X= 2(2 - 1/3) = 0.54-, ^= :i(2+ Vs) =7.5. 

Applying the method § 138, we have 



/<Py 



(x^ + 4)^ 



-3 [^ - 0-54][-^ - 7-5] = - 0.2. 



(3)....- (-"fc.['+*-"i -+•■'■ 



- [x + 2][^ ~ 0.54] = - .0013. 



,dx^]^=,,, {x'^4r 

Hence, (— 2, — i), (0.54 — , — 0.05), and (7.5 — , 2.6 — ) are points 
of inflexion. 

2. ;p — 2 — s(x — 2)3/5. 
dy _ - 9 



dx 5(x - 2)^5* 
d-^y 18 



dx^ 25(x - 2)V»' 

18 

~~r ^T7^ = o has no real 

25(;c — 2)V5 

finite roots. 

18 

25(x — 2)V5 



M 



00 gives X = 2, 



{dy/dx%<2 is negative, (dy/dx%>2 is positive. 
Hence, jt = 2 = ^ is a point of inflexion. 



2"]^ DIFFERENTIAL CALCULUS. 



3- y — ^^Vi^ — x)lx. , X — 3^/4 

4. a^y = (x — by. X ^= b. 

5. y{x — 2) = {x — l){x — 3). X = 2. 

6. a'^y = x^. X =: o ^= y. 

7. y'^(x — a) = x^ -\- ax'K x =■ — 2a. 

8. J = x'^/a -\- a\{x — a)/dfl^. x = a. 

g. y = x"^ log (l — x). X = O ~ y. 

10. x'^y — 4a^{2a — y). x = ± 2a/ ^'i. 

11. y = <?1A. X = — 1/2. 

12. x^ -\- y^ ^= a^. X =^ a, x = O. 

13. logy = i^x. ^ = 8. 

14. jj/ = x^ tan X. X ^= o ^= y. 

15. y = a sin (x/b). x = o, bit, etc. 

16. jj/ = (« — jr)V2 + ax, X = a. 

I"], y = x"^ — x^/^. X — 64/225. 

1/3 

\?>. y = e^ ^ = 8. 

19. y = x^/{a^ + ^V' X = o, jr =: ±<3!|/3. 

20. J = a tan {x/b). x = o = r. 

21. 1/ = x'^{x -f- <'7)/[a(x — a)\. X = — a[\/2 — i)- 

22. y^ = ay^ — x^. X = a. 

23. J = 2 -|- (x — 2)^. :r — 2. 

24. y = axy -[- by"^ -\- cx3. x := o = V- 

25. jf(«* — '^*) = ^{■x — «)* — •^^*. X = ^^/s, X = a 

26. y = a^xlixP' -(- a'^). x = o, x — ±a\/2. 

27. 7 =: x'^/a — x^y/a^. x — ± a/i/3. 

28. J = X cos (x/a). X = Q = y. 

2g. y = X -\- 2^x^ — 2x3 — X*. X = 2, X = — 3. 

30. / = a\/x/{2a — x). X = ^/2. 

31. axy = r8 — a^. x = a. 

32. a'^jK — -^^/S — dX"^ -\- 2^5. X = rt, 

33. xy = a^ log {x/a). x = ae^/^. 
34 {ay — x2)* = (5^:^ X = gb/64. 

35. aV ^ ^'(«' - ^^)- X = ± a/s/l. 

36. yix'' f «') .-= ^"^(^ - x). 

37. ;t V - ^^'(-^' - J'^)- X = o. 



SINGULAR POINTS. 



277 



38 


y = 


2ai/{2a 


-x)/x 


39 


y ~ 


te-^^f^^^ 




40 


a^j 


= 3^j;2 - 


- x\ 


41 


y = 


xe''. 




42 


«|/^ 


= {x - 


a)\/y. 


43 


;f^ = 


e-V^. 




44 


a-^y: 


= x(a2 — 


x'^). 



X = 3«/2, j(/= ± 2a /\/ 2,. 



X = a\/ {n — i)/n. 

X = b 

X = — 2. 

a' = — 2a. 

X = 1/2. 

X = o = J. 

162. Polar Coordinates. — Having i'=f{6), it follows 
from § 160 that r = c corresponds to a point of inflexion if 
dp/dr = (u -}- d"^ u / dB'^^f / r"^ changes its sign in passing 
through {dp/d7')r^g. 

Hence, the real roots of the equations dp/dr = o and 
dp/dr = CO, or, what is equivalent. // + d'^u/dO'^ = o and 
u -|- d'^ii / dd"' = 00 , are critical values which may be lested 
by methods similar to those indicated in §§ 135 and 136. 

It should be observed that/, corresponding to a point of 
inflexion, is a maximum or a minimum. 

EXAMPLES 

I. r = a6y{e' - i) = i/u. 

Whence du/dO = 2/{a6'), d''u/d6' = - 6/(ae'). 

u + d''u/d6'' = {6' - 6' - 6)/{a6') = o 
gives 6^= db V3, and changes sign as passes through either. 
Hence, B= ± V ^, r — 3^/2, are points of inflexion. 
'• 2. r= (a-\- a6)/6 = \/u. 
Hence, du/dB — a/{a -}- adf^ 
d'u/dd"- = - 2a'/(a + ad)\ 

d'u_a '{d'+2e'+e-2 ) _ 
^~^d6'~ (a + aey ' -"" 

gives 6 = 0.']—, and dp/d^ 
changes from — to + as 
passes through 0.7 — . 

3. r = ae-y\ d=jj2 




278 



DIFFERENTIAL CALCULUS, 



163. A Multiple Point is a point common to two or more 
branches of a curve, and is double or triple^ etc., according 
to the number of branches. They are classed into Points 
of Intersection^ Shooting Points^ and Points of Tangency. 

A Multiple Point of Inter- 
section is one through which 
the branches pass and have 
different tangents. ^ is a double 
and <^ is a triple point of inter- 
section. A double point of in- 
tersection is also called a node. 
A Shooting Point is a multiple 
point at which the branches terminate with different 
tangents. 



A Salient Point is a double shooting 
point. 






A Multiple Point of Tangency* 
is one at which the branches have 
a common tangent. 



A Cusp is a double multiple point of tangency at which 
both branches terminate. 

When in the vicinity of a cusp the branches are on 
opposite sides of the common tangent (Fig. c\ it is of the 




* Sometimes called points of osculation or double cusps. 



SINGULAR POINTS. 2/9 

first sfecies, or a keratoid (horn) cusp; when on the same 




side (Fig. ^), it is of the second species, or a ramphoid 
(beak) cusp. 

A Conjugate Point is an isolated real point of a curve. 
Thus, the origin ^ = o=j^' is a real point of the curve 
whose equation \-> y = ± x Vx - 2, but y is imaginary for 
± X < 2. Hence the origin is a conjugate point. 

Let J = ^{^) be the equation of any curve. 

From § 124, 

jr(c -{- /i) = F{c) + F'{c)h + F"(c)hy2 + etc. 

When {c, d) is a conjugate point, Fi^c -\- h) is imaginary 
for values of h near zero, while F{c) and h are real. Hence, 
one or more of the expressions F\c), F"(c), etc., must be 
imaginary. This important characteristic of a conjugate 
point is frequently used in testing critical points. Thus, 
(r, d) is a conjugate point provided F{c) is real, and F"{c) 
is imaginary for any entire value of ;/. 

In the example above we find F'{o) = -^ V — 2. 

Since the ordinates of points of a curve adjacent to a 
conjugate point are imaginary, the number of such ordinates 
for each point is even. It follows that a conjugate point is 
a multiple point in the immediate vicinity of which the 
branches are imaginary. The tangents corresponding to a 
conjugate point may be real or imaginary, coincident or 
separate. 



28o 



DIFFERENTIAL CALCULUS. 



Having the equation of any curve with two or more 
branches, if either variable, as y^ has but one real value, 
d^ corresponding to any real value of the other, as x = c^ 
{c, d) is a critical point. 

If {dy/dx)(c,d} has two or more values, {c, d) is a multiple 
point of intersection or tangency according as the several 
values of {dy/dx)(^c, d) are unequal or equal. 

When the values oi y for points adjacent to and on both 
sides of {Cy d) are imaginary, {c, d) is a conjugate point. 

EXAMPLES. 



I. ^ = 3 ± (^ — 4) '^•^ — 2. 




^ < 2, ;; is imaginary, j*; =: 2, j = 3. 
2 < jc < 4, jv has two real values. -^ = 4, ^ = 3. 
X > 4, y has two real values. 
Hence, (2, 3) and (4, 3) are critical. 

dy/dx = ± Vx — 2 ± {x — 4)7(2 Vx — 2). 
{dy/dx)(fi,z) =00. (2, 3) is not a multiple point. 

Hence, (4, 3) is a double multiple point of intersection. 

,. 2.. ;;,= ± Vx\x - ~2)/ V3. 

x^=^o-=y. _y is imaginary when ^<o, or o<Ji:<2. 
Hence, the origin is a conjugate point at which we have 

{dyldx),= -^ 2I Sf^l. 



SINGULAR POINTS. 



281 



X = o=y. X = o ± h^ y has three 
values, (dy/dx)^ = o and ± y/ 2. 




Hence, the origin is a triple point of intersection. 

Y 4. J^'= 2 -f- .T tan~^(i/x) = 2+x cot~^jc. 

^=: o, J)/ = 2. When cot~^Jc<7r/2, 
X and cot"^^ have the same sign, and ±.x 
give equal positive value for j. 

{dy/dx),= [cot-^^-V(i+^')]o = ^/2 
and z^/2 and 57r/2, etc., or— ;r/2 and 
— 37r/2 and — 57r/2, etc., according as 
X ^B-»o is positive or negative. Hence, 
(o, 2) is a shooting point. 

d'^y/dx^ = — 2/(1 + jr'^)' is negative for ± ^; hence, the 
curve is concave downward. 
5. j = V(i+^'/"). 
:xr = o = J. 







= o or I 

according as ^^->o is positive or negative. Hence, the 
origin is a salient point. 

Y 6. ay = cx'-\-x\ 




y=±.x''Vc-\- x/a, 

X =^ o =y, and ji^ has double 
values for values of x between ± c 
and o. 



\dx^o i2ai^X']-cJo 



282 DIFFERENTIAL CALCULUS. 

Hence the origin is a dpuble point of tangency. 
7. y = '^±{x — 2fJK 




dy/dx = ± 3 \x — 2/2. 
x<2^ y is imaginary. 
x'>2^y has double values. 

x=2,y=i, and (dy/dx\^^^)— ±0. 

Hence, (2, 3) is a cusp. 

The branches are on opposite sides of the common tan- 
gent JF = 3, and the cusp is of the first species. 

8. (2x-j)' = (x-3)'. 

^<3 makes 7 imaginary, ^=3 givesj)^=6, and ^>3 gives 
real double values to j. 

(dy/dx)(^^^) = 2 ± o. Hence, (3, 6) is a cusp. 

Since ji^ = 2^ ± (a: — 3)^2 ^ the branches are on opposite 
sides of the common tangent j' = 2x^ and the cusp is of the 
first species. 

A characteristic of a cusp of ihe first species is a change 
in direction of curvature from one side of the common tan- 
gent to the other ; while at one of the second species the 
direction of curvature remains upon the same side of the 
common tangent. Hence, different signs for d'^y/dx^ cor- 
responding to the two real values of y in the immediate 
vicinity of a cusp indicate the first species, and like signs 
the second species. 

Thus, in example (7), (^-oh= (± — has val- 

ues with different signs. 

In some cases it is preferable to consider J^' as the inde- 
pendent variable. 



SINGULAR POINTS. 



283 




g. jj; = 3 -|- (jt: — 2)^/^. y has but one 
value for each value oi x. x =^ 2,y ^= ^, 
dy/dx =2/[3(jt: —2)^/^] is negative when 
^>2, is 00 when x ■=■ 2^ and is positive 
when x<2, (2, 3) is evidently a cusp of 
the first species as in figure. 

Solving with respect to x^ we have 

x=2±[y- 3)V2 , {dx/dy\,^ 2) = ( ± 3 '^^F^z/Az, ?)= ± o. 
j<3, ^ is imaginary. J = 3, ^'c = 2. j>3 gives double 
value for x. Hence, (3, 2) is a cusp. 

|— 7^ ] = ( ± ,/ I has values with different signs, 

therefore the cusp is of the first species. 

Points corresponding to maximum or minimum ordinates 

at which dy/dx = 00 are cusps. 

5/2 




10. y^=-x'^-^x 
~=2x±sxyy2. 

dx"" 



2±i5^vy4. 



j; = o = jj/. .^c < o, ^ is imaginary. 

x>Oy y has two real values. 

(dy/dx)^ = ± o. Hence, the origin is a cusp at which 
the axis X is the common tangent. 

For o<:x:<i both values of y are positive; therefore the 
cusp is of the second species. This is also indicated by the 
fact that both values of [d'^y / dx'^\<:^x<^^/%%^ are positive. 

Examine the following curves for multiple points: 

11. y^ — x^/{ia — x). (o, o) is a cusp of I St species. 

12. (y — x)2 = x^. (o, o) is a cusp. 



284 DIFFERENTIAL CALCULUS. 



13. jj/ = ± {x'^^x^ — 4)/4. jc = o =_j/ is a conjugate point, 

14. (4/ — x^Y — (•^— 4)^(^ — 3)*. (3, 27/4) is a conjugate point. 

15. j/^ = 2x^ -^x^. (o, o) is a double point of inter- 

section. 

16. x^t^ •\- yV^ ■=■ (^l\ (o, ± a) and {± a, <S) are cusps 

of ist species. 

17. ^5 — _y4 — :t:«, (o, o) is a double point of tan- 

gency. 

18. {y^ --c^f = x\2.X'\-'iiaf, (— 3a/2, ±d) are cusps of ist 

species. 

19. y^{cfl — x^) = ^«. (o, o) is a double point of tan- 

gency. 



20. J = ± ^4/1 — jf*. (o, o) is a double point of inter- 

section. 

21. (;c — a)* = ()' — :v)>. (d!, a) is a cusp of ist species. 

22. ojj/" = x^, (o, o) is a cusp of ist species. 
12,, y "= a. -\- x ■\- bx^ ± co^l^. (o, a) is a cusp of 2d species. 

24. y^ = «'jc2 — jf*. (o, o) is a double point of inter- 

section. 

25. O' — ^ — ^^*)^ = (^ — of. (a, ^ + ^^'^) is a cusp of 2d species. 

26. jj/=±;c[4/«*+-*" / 4^^^"~-^^]« (Oj o) is a double point of inter- 

section. 

27. (y — ^)^ = (^ — a)6. («, /J) is a cusp of ist species. 

28. (jy/+i)'^+(-^~i)^('^— 2)=o. (i, — I) is a cusp. 

29. ;j/3 -|- jt^ = 2ffx' (o, o) is a cusp of 1st species. 

164., Let u = f{x^y) = o be the equation in a rational 
integral form of any algebraic curve, then (2) (§ iii), 

^ _ _ 9^ /9f^ , . 

dx~ dx'dy ^'^ 



SINGULAR POINTS, 28$ 

At a multiple point dy/dx has two or more equal or un- 
equal values. 

Since dufdx and 'du/'dy are rational integral functions, 
each can have but one value for any set of values of x 
and y. 

Hence, equation (i), two or more values of dy/dx require 

"du/dx = o and d^/dy = o (2) 

Any set of real roots of these equations as (r, ^), which 
also ssitisiy /{Xf y) = o, are therefore critical for multiple 
points. 

(dy/dx)(c,d) may be evaluated as in § 117, otherwise (3) 
{§111) gives 

dx''^^dxdydx'^dy\dx) -<'>••• V3) 

from which the two values of (dy/dx)^c, d) may be found. 

If in (3) d"u/dx\ d"u/dx dy and d'u/dy^ vanish for 
(Cf d)j (dy/dx) is indeterminate. Then (7) (§ in), 

a^' "*" ^-dx'Zy dx "^ ^-dx-df \dxl "^ df \dxl " ^' ^^^ 

gives three values for (dy / dx) (c, S). 

It follows that any algebraic curve whose equation in a 
rational integral form contains no term of a degree less 
than the second, with respect to the variables, has a mul- 
tiple point at the origin. 

EXAMPLES. 

I. u= y"^ — x*(i — x"^) = o. 

^u/dy = 2y = o, du/dx = — 2^(1 — x*) + 2x' = o, give 
X —y = o. 



286 DIFFERENTIAL CALCULUS. 

Hence, the origin is critical. {3) gives {dy/dx)^ = ± i. 
y has double values for + 1 > ^ > — i, hence the origin is 
a double multiple point of intersection. 

2. u — x^ -^ xy — y = 0, ] 

du/dx = 4x^ -f 2xy = o, r give x = o =y. 
du/dy = x' - 3/ = o, J 

'd'u/'dx'' = i2:r-^ 4- 27, a'V9-^9j = 2^. a'^v'9/ = - 6r, 

-d'u/dx' = 24:^, a' V9^'9j^ == 2, 9'2^/ajt: a/ = o, 

dW^/ = -6. 

From (4), (dy/dx)^ — o, and ± i. Hence, the origin is 
a triple point. 

3. ti = (4y - 2,xY - {x- 2Y/2 = o, 
di^/dx — — 24y-{-24X — ^x'^/2 — 6=0, 
du/dy = S2y — 24X = o, 

a'2/t/a^' = — 3"^ + 24, d^u/dxdy = — 24, a'V9/=32. 

Hence, {dy/dx)(^.2, 3/2) = 3/4 ± o, 

since j^ = [3^ ± |/(:\: — 2)72 ]/4. 

^ < 2,_y is imaginary, x > 2,y has double values. 

Since x > 2 gives one value of y greater and the other 
less than sx/4, the two branches are on opposite sides of 
the common tangent y = 3X/4. Hence (2, 3/2) is a cusp 
of the first species. 

4. z^ =: y"^ — x^ — 2ax^ — a^x = o. 
dt^/dx = — ;^x^ — 4ax — d^ = o. 
du/dy = 2y = o._ 



give X = 2f 

y = 3/2- 



SINGULAR POINTS. 28/ 

Hence, (— ^, o) is critical. From (i), 






o 



<§"'^ &i?L,o,=( 



6-+4'^\ =±|/^. 



2dy/dx /(-a,o) 

Hence, (— «, o) is a conjugate point. 

5. u = y'^ — 2x'^y — x*y + 2x^ = o. 
'da/'dx ■= — 4xy — ^x^y -\- 8x'^ = o. 
'du/'dy = 2y — 2x^ — x" — o. 

Hence jc = o =^ is a critical point. 
From the equation of the curve, 

y^i^x'-Y x*/2) ± x' i/- 4 + 4x' + xi2, 

and is imaginary when x is near zero. Hence the origin is 
a conjugate point. 

Examine the following curves for multiple points : 

6. x^ — 2)0'Xy-\-y^ = o. (o, o) is a double point of inter- 

section, 

7. x*—2ay^—2>^^y'^ — 2a'^x'^-\-a*=o. (o, —a) and {±. a, 6) are double 

points of intersection. 

8. X* — 2ax'^y — axy''' -\- a^y'^ = o. (o, o) is a cusp of 2d species, 
g. jj/* — axy^ = — x*. (o, o) is a cusp of ist species. 

10. {x^-\-y^y = a'^{x'^ — y"^)' (o, o) is a double point of inter- 

section. 

11. ay^ -f- dx^ = x^. (o, o) is a conjugate point. 

12. ay"^ — x^-\-4ax^ — c^a'^x-\-2a^ =0. (a, o) is a conjugate point, 

13. X* — ax^y -\- axyi -\- a-^y^ = o. (o, o) is a conjugate point. 

14. y^- = x(x -\- ay. (— a, o) is a conjugate point. 



288 DIFFERENTIAL CALCULUS. 

15. {y — 2Y = {x — \y{x — 3). (i, 2) is a conjugate point. 

16. y\x^ — a^) — X*. (o, o) is a conjugate point. 

17. x'^ -}- x^y^ — tax^y -\- a'^y'^ = O. (0,0) is a double point of tan- 

gency. 

18. a^y^ — 2abx^y = x^. (o, o) is a double point of tan- 

gency. 

19. x* — axy'^ = ay^. (o, o) is a triple point and a cusp. 

165. A Terminating Point {^oinf d' arret) is one at 
which a single branch of a curve terminates. 

EXAMPLES. 

\. y =. X log X, 

^ = o = jl^. y is real when jc > o, and is imaginary 
when X < o. Hence, the origin is a terminating point. 

2. y = ^~V^. 

As + Jt: ^>-^ o, y ^^ o, and as — ^ »»-» o, y ^-^ 00 . Hence, 
the origin is a terminating point for the right-hand branch. 

3. x^ log X -{-y = xy. (o, o) is a terminating point. 



CURVATURE OF CURVES, 



2^9 



CHAPTER XV. 



CURVATURE OF CURVES. 



PLANE CURVES. 

1 66. The Total Curvature of an Arc of Any Curve is 

the angle which measures the change in direction of the motion 
of the generating point while generating the arc. 

Let MM' = ds be the length of any varying arc not in- 
cluding a singular point, of any curve. At M and M\ 
respectively, draw the tangents 
MT and M'T\ Each tan- 
gent indicates the direction of 
the motion of the generating 
point corresponding to its 
point of tangency. The angle 
TRT\ denoted by (^^, in- 
cluded between the tangents 
at the ends of the arc, measures the change in direction of 
the motion of the generating point while generating the 
arc Ss, and is its total curvature. 

If the extremities of an arc coincide, forming a closed 
curve without singular points, the corresponding tangents 
coincide, but the total curvature is 2 tt and not zero. 

167. The Rate of Curvature of a Curve at a Point is 
the rate of change^ at the point, of its direction regarded as a 
function of its length. Thus, in the preceding figure, let ^ 




2Q0 



DIFFERENTIAL CALCULUS. 



represent the angle which the tangent MT makes with X. 
It determines the direction, with respect to X, of the 
motion of the generating point at M^ and, regarding ^ as a 
function of the length of any varying arc of the curve, as 



AM 



dip _ limit 



''di = 



Ss M-^ o 



67 



is the ra^e of curvature of 



the curve i- at J/". (§70.) 
168. Rate of Curvature of a Circle at a Point.— Let C 

be the centre and r the radius of any circle. Then 

ds 8s:B-^o\_dsJ \_rdipj r 




Hence, in any circle the rate of curvature is the same at 
all points, and at any point is equal to the reciprocal of its 
radius. 

169. Circle and Radius of Curvature — When the radius 
of a varying circle decreases continuously^ the rate of cur- 
vature of the circle at any point increases continuously. 
Hence, a circle may always be assumed having at all points 
the same rate of curvature as that of any given curve at any 
assumed point. 

Such a circle tangent to the curve at the point assumed, 
and having the same direction of curvature, is called the 



CURVATURE OF CURVES. 29 1 

circle of curvature of the pointy and its radius and centre are 
called, respectively, the radius and centre of curvature. It 
follows that a radius of curvature is normal to the curve. 

Any chord of a circle of curvature which passes through 
the point of tangency is called a chord of curvature. 

170. Curvature of a Curve at a Point. — Representing 
the radius of curvature at any point of any curve by p, we 
have 

dtl)/ds = i/p (i) 

That is, the rate of curvature of any curve at any point is 
equal to the reciprocal of the corresponding radius of cur- 
vature, and the rates of curvature at different points are in- 
versely as the corresponding radii of curvature. 

dtp/ds, corresponding to any point of a curve, multiplied 
by the unit of length of s is (§ 68) the change that the cor- 
responding value of ^ would undergo were it to retain its 
rate at the point over the unit of length of s. In other 
words, dip/ds multiplied by the unit of s is the total curvature 
of a unit of length of the corresponding circle of curvature, 
and it is generally called the curvature of the curve at the 
point or the curvature of the corresponding circle of curvature. 
Its numerical value is the same as that of the correponding 
rate of curvature, and for reasons similar to those given in 
§ 95 it is generally used instead of the rate. 

It is important not to confound this curvature of a circle, 
which measures the so-called curvature of the curve atapoifit, 
with the total curvature of an arc described in § 166. 

f) and s must be expressed in terms of the same unit of 
length, and at any point where p = unit of length the rate 
of curvature is unity, and the corresponding curvature of 
the curve is a radian, which is therefore the unit of cur- 
vature. 



292 



DIFFERENTIAL CALCULUS, 



EXPRESSIONS FOR RADIUS OF CURVATURE. 

171. From (i) (§170), p=:ds/dip (i) 

which enables us to determine the rate of curvature at any 
point from the equation * of the curve in terms of s and ^. 
Thus, having s = c tan ^; for a catenary, 

i/p=(cos'^)A = ^/{/ + ^'). 

2. s = ctp p = c. 

3. s = asimp p = a cos ^. 

172. 'Lety=f(x) be the equation of any plane curve. 
Having, as before, AM = j, let 

T^ />/" = dx. Then dx = cos ipds, 
dy — sin i^'ds, tp = tan~^(^/^A:), 

dx' 




dtp 
^ dx 



i=»'.+(i)' 



or 



ds. ~ ds/dx ~ dxy L' ^dx) Jdx 

P = [i+/wTV/"W (i) 



from which the radius, and therefore the rate of curvature, 
may be found from the equation of the curve in rectangular 

coordinates. 

Adopting the positive value of [i +/'(-^)'']'^'>Pwill have 
the same sign as/"(^), which determines the direction of 
curvature. 



* Called the intrinsic equation of the curve. 



CURVATURE OF CURVES, 293 

In general, at a point of inflexion (§ 161) 
f"{x) = o, or 00. 

Hence, at such a point p is generally 00 or o. In general, 
at a multiple-point, /'(^) has two or more different values;, 
hence, p has two or more values, one for each branch. At 
a multiple-point of tangency, f'(x) has but one value, but 
f"{x) has, in general, a value for each branch; hence, p dif- 
fers for each branch. 

Comparing (i) with (3) (§ 139), we see that the centre of 
curvature corresponding to any point of a given curve coin- 
cides with the point {x^y) (§ 139) whose distance from the 
curve measured along the normal is, in general, neither a 
maximum nor a minimum, and whose coordinates are (2) 
(§ 139) 

It follows (§ 139) that the circle of curvature correspond- 
ing to any point of a curve, in general, intersects the curve 
at their common point of tangency. 

This is not the case, however, at any point where the 
curvature is a maximum or a minimum, which, in general, 
includes any point in the vicinity of which a curve is sym- 
metrical with respect to the corresponding normal. At such 
a point the curvature, in general, decreases or increases in 
both directions; consequently in that vicinity the circle of 
curvature is interior or exterior to the curve. 

To illustrate, take the ellipse 

ay + b'^x' = a'b\ 

f(xf = b*xy{ay\ f"{x) = - ^v(^y). 



294 



DIFFERENTIAL CALCULUS, 



Hence, i/p = abl{a' -\- b"" - x" -/)'/» 

= - a'b'l{ay^ b'xj/^ (3) 

At the vertices (±^, o) i/p = a/b"^^ a maximum. 

At the vertices (o, ± b) i/p = /^/V, a minimum. 

Hence, at the vertices of the transverse axis the circle of 
curvature is within the curve, at those of the conjugate axis 
it is outside, and at all other points it cuts the ellipse. 




Also (2), x=:x-x{ay-^rb'x^)/{a'b'') = (a'-'b^)x'/a''; 



(4) 



y-y-y{ay-^b'x'')/{a'b') = - {a^- b')yyb\ 

In (3) put/=-^'^ (a^^x')/{a\ and {a'-b')/a'=e\ whence 
b~a i/i^^'. Then 



i/p=^:aWi-e'/(ci'-e'xy/'. . . • (5) 



EXAMPLES. 



I.' y=2X-\-$. 



cf7~\'^ 



/W=4, /V) = o, p = «), 



2. x''-[-y = r\ 

Jycf = xV(r^ -~x% fix) = rV(r»-^»)3/8, 

_/ ^^y/s / r' _ 



CURVATURE OF CURVES. 295 

3. yi = 2pX. 

^~y~^2x) / (2/^)3/3- J^ • 

Px=o = p — one half the parameter. 

4. ity^ + 4-^' = 64. 

From (3) and (4) (§ 172) we have at (2, 4/3) 

p = - 5.86, 7= 3/8, 7= - 9I/3/4. 

5. //^'-^ - ^Va- = - I. 

_ {a^x'-y b'^x'' - a'^fl'^ _ (ay -\~ d^x^f/^ _ {e'x' - a'fl^ 
^^ a^b ~ a'b'' ~ ab 

X = ± a, y = O, and p = b^/a, 

a=b, p = {2x^ - ay/ya\ 

6. ;jr« = 2px + r'^xK 

_ \2px + r'x'' + (/ + r^-y)"]^/^ 
/'^ 
Hence (Example 3, § 149), at any point of a conic, as given, p is 
equal to the cube of the corresponding normal divided by the square 
of half the parameter. 



T. X ■=■ r vers-i (y/r) — \2ry — y^, 

P = - [l + {'^r/y - l)]3/2/(r/y) = - 2 '\f2^. 

Hence (Example 6, § 149), at any point of a cycloid p is equal to 
twice the corresponding normal. 

y ■=. Q) gives p = o, and y •= ir gives p = 4n 

8. >r = 4 — 3(x — 2)3/5. 

(2, 4) is a point ot inflexion at which p ■= o. Ex. 2, § i6r. 



296 DIFFERENTIAL CALCULUS. 

9. y = 9^. At(3,i/27) ^= 13.5, 7= - VS 

10. ;j/2 = 8;tr. /3 = 2{x + 2)3/2/1/2. 

11. ^79 + ;/V4 = I. /3^=o = 9/2. 

12. ;fj/ = /«. p = (^'^ -\-yy/^/2m, 

13. jj/ = a(<fV« + ^-*/«)/2. p = /A, 



Jir 



14. Jr2/3 +//3 = a^A p = 3 t^a^ 

•^ = -^ + 3 |/-^/* y=y + 3 Vx'^y' 

15. 3aV = x\ p = (a* + xy/^l^a^x, 

16. _y = ^3 — jf' -|- I. /j^^jyg = 00 (point of inflexion). 

17. ^A = sec (^p/a). Q •=. a sec (jf/a). 

18. jj/" = 6^2 ^ ^3. p = - [/ + (4^ + x^Y^y^x^y, 

19. jj/ = jf4 _ 4jf3 — i8jf2. p^^^ = _ 1/36, p;c=3 = «> . 

20. y = ae'^/a, p = (a» -\- y'^)^l'^/ay. 

21. y = fl'jf. j^ = (o4 -j- i5jf/4)/6)/a'», ^ = (a*^ — 9y^)/2a*, 

p = (9;/4 -j- aif/z/tay. 

22. 3ay = 2X\ p = ± (2« + 3Jf)V2 |/jp/« ^3. 

23. jf = sec 2y. p = (2;*:' — iy/4x. 

24. ;/ = loga^. p = (iJ/a' + X^)3/2/MaX, 

pa=e = 2 i/2. 

25. ;/ = jf» - jf« + I. p^=o = - 1/2, /Oy=„^. = 1/2. 

26. ^x-\-^y- \/a. p = 2{x +yf/y Va. 

27. ay"" = x\ p = i4a + ^xf/'^xV^Jta. 

28. jj/' = ;»;V(2« — ;tr). p = a(8a — 2>x)^/2xyy 2t{'^a — ;i;)'. 

29. «^/ = 3x^ -{- cx^y. /3o= CO. 

30. J = log sec X. p = sec x. 



CURVATURE OF CURVES. 



297 



31. In a parabola show that any radius of curvature is twice the 
part of the normal intercepted between the curve and the directrix. 

N.P. 

32. Applying (5) to a meridian y 
of the earth, we have, since 

/ •=■ latitude = ^ — ^/2, 
tan' / = cot'^ ^ = i/TX^)'. 
For an ellipse we have 

tan" / = a\a^ - x'')/b''x'' = (a^ ~ ^')/(i - ^)x\ 




Hence, x^ 



I + (i - e") tan^ / ~ sec' / - / tan' /' 



, , 5 2 ^HsecV-^'sec'/) a\i - e") ^ 

and c^ — e^x" — ^ , ^ ^ ^ = — ^ • -f ; , 

sec' / — / tan' / \ — e^ sinV 

which, substituted in (5), gives 

i/p = (i - ^» sin' /)3/V^(i - ^0- • 



(5) 




173. In Polar Coordinates — Differentiating, 
X •=. r cos B, y ^= r sin ^, 
and substituting in (i) (§ 172), as in Example 11 (§ 115). 



we have 



-[''+S"]7["+"S)'-'S]- (■) 



Rice and Johnson's Calculus, p. 355. 



298 • DIFFERENTIAL CALCULUS. 

Putting r — i/u, whence dr/dd = — {i/u'^){dii/dd)^ and 
d-'r/de' = {2/u'){du/d6y - (ilic'){d''u/dd\ and substitut- 
ing, we have 

^=[--+g]7[-'(»+s)]- ■ • • w 

EXAMPLES. 

1. r = ae. P = a(l+62)3/2/(2-j-e2)^ (a'-fr2)3/2/(2a»+ r2\ 

2. r = a^. p = r 4/1 + log* «. 

3. r = asinnB. Pr=o^^ ^^/^' 

4. r —2a cos 6 — iz. p = ^(5 — 4 cos 6)^2/(9 — 6 cos G). 

5. ?- = «(i — ^^)/(i — e cos 6). 

p = «(i - ^2)(i _ 2^ cos 6 + ^'^)VV(i — ^cos 9)3. 

6. /''^ cos 20 = «2. p = — rYa^. 

7. r = a(2 cos — i). p = ^(5 — 4 cos 0)V2/(g — 6 cos 0). 

8. /- COS^ (9/2) =/. p = 2^3/2/ V/. 

9. r = fl! sec'^ (0/2). p = 2a sec^ (0/2). 

10. r = a0-*. p = K4«*+^')^/V2«'(4«^ — i^*). 

11. r'Bz=za'^. p = ri4a^ -\-r^)y^/2a\4a^ - r% 

12. r = 4a sin^ (0/2). 
From which and § 150, 

dr/{raB) = cot (0/2) = cot 0. .*. 0/2 = <p. 
^ = 0-1-0 = 30/2, and dip = 3de/2. 
Therefore, 
p = ds/dip = {2/3){ds/de) = (2/3)?- cosec = (8/3 )« sin (0/2). 

13. rQ = a. p = r(a^ + r^f/ya^ = a(i -\- Q^f/yQ\ 

14. r^ = d^ cos 20. p = aVsr. 

174. From § 172 we have 

dx = cos tpds, and ^ = sin tpds. 

Differentiating without assuming the independent vari- 
able, and writing ds/p for dip^ we have 

d^'x = cos r/^'d^s — sin t/:{dsY/p, 
dy = sin ipd^s + cos tp{dsy/p. 



CURVATURE OF CURVES. 299 

Squaring and adding, we obtain 



Hence, p = ds'' / V {_d^' x)' + {d^yf - (^V)^ . . . (i) 

which is a general expression for p. 

Regarding s as the independent variable d'^s = o, and (i) 



becomes p = ds'/V{d'xy + (^»% 

from which i/p' = (d'x/dsy + {dy/ds'y, ... (2) 

which is a convenient form when x and j are given as func- 
tions of s. 

175. In (i) (§ 172), ^ is the independent variable. Sub- 
stituting (dxdy —dyd''x)/dx' for d'^y/dx^^ we have (§ 115, 
Example 11) 

p = (dx" -f dyy/y{dx dy - dy d'x), . . (i) 

in which neither x nor ^ is independent. This form is con- 
venient when X and y are given as functions of a third vari- 
able. Thus, having 

I. x = a(0 — sin 0), y = a{i — cos 0), 

we have, as in Example 12 (§ 115), p = — 4^ sin (<p/2), 

!x = c sin 26(1 + COS 29). 
jj/ = iT cos 26(1 — cos 20). 



p = 4<r cos 39. 



I y = 6 sxn B. 

!x = a{2 cos 9 -{- cos 29). 
y = a{2 sin 9 - sin 29). p = - Sa sin (39/2). 

{ x = {a + 6)cosQ - d cos ^-4^9. 

5. i , . ^ = , / sin -9. 

J/ = (a + ^) sin 9 — ^ sin — ! — 9. 



300 DIFFERENTIAL CALCULUS. 

176. p in Terms of r and p.— From § 150. 
Hence, 



dp_ 
dr 



=[''+-(S)'-''»]/[''+sr- 




Comparing with (i) (§ 173), 
we have 



p = rdr/dp. 



(0 



Let C be the centre, and 
CM = p, the radius of cur- 
vature corresponding to M. 
Let c = MN represent the 
chord of curvature which 
passes through P, and PM = r. Then 

c=2p cos PMC = 2P sin OMP = 2pj>/r = 2pdr/dp. 

EXAMPLES. 

1. p:=ar, p = r/a, c — ir. 

2. ar =p\ p = ip^ld" = 2(rVa)i/a, c = 2^V« =2r. 

3. r^ = ap. p = a/2, c — r, 

4. a> _|_ 32 _ y3 = a23V/>. p = a2^Vi>^ 

5. r^=d^p, p = a^/sr, c = 2^/3. 

6. rS = 2fl!/^ <^ = 4^/3- 

7. r'w+i =«»«/. /3 = ^V(^^ + i)/» '^ = 2r/(w + i). 



8. f = r^ — «2. 



A>=A 



2f/r. 



CURVAI URE OF CURVES, 



301 



9. r^ = «2 cos 29. 

Putting ?*-* = «, we have a^u^ = sec 20. 

Hence, du/dB = u tan 20, m^ _|_ ^^^2/^92 = a^u^ i// = a*/r\ and 
<^/^r = z^'^/a^. Therefore, p = a'/sr, and c = 2^/3. 

10. ;' = 2a{i — cos 0). ft = Sa sin (0/2)/3 — 4V«V3, <^ = WS- 

11. r = a(i + cos 0). p = 24/2^^/3, ^ = 4,V3- 

12. ;- = aB. 

dr[dB = a; also (§ 150), ^V'^'^ = rj/r^ -/'//. 
Hence, r\^r^ -p'^/p = a, or r* - ^V^ = a^'. 

From which ^^r/^ = /(r=^ + flt«)/2(2r'' - /). 

But /' = rV(^' + «'). Therefore, p = (r» + a»)3/2/2(r' + 2a»). 
The chord of curvature through the pole 

= 2pdr/dp = 2pHr'^ + a^)/2r{2r^ -p^) = r(r'^ + a»)/(^« + 2a'). 
Hence, /O = ^ when (^r' + ^2)3/2 = 2r(r2 + a') or ?- = a/4/3. 

CURVES OF DOUBLE CURVATURE. 

177.* Let MM^ = Ss be the length of any varying arc 
not including a singular point, of any curve of double 
curvature. At M and J/' re- 
spectively draw the tangents 
MT and M' T\ and through 
the origin draw the two right 
lines OE and OB^ parallel to 
them respectively. Upon each 
lay off a length /, and join the 
extremities jS and jE' by the 
right line EE\ forming an 
isosceles triangle in which the 

angle ^6>^', designated by St/;, measures the fofal cnrva.- 
ture of the arc Ss. (§ 166.) 

* Modification of method in Calcul Differentiel, par J. Bertrand, 
page 614. 




302 DIFFERENTIAL CALCULUS. 

The rate of curvature of the curve at M is (§ 167) 



ds 



= ,"""' m (X) 



Let «', A r* ^^^d a + 6a^ ft -[- ^y5, ;/ + dy, represent the 
angles which the tangents MT and M' T' make respec- 
tively with the coordinate axes. 

From the triangle EOE' we have 



EE' /l= 2 sin (<^^/2); hence, 



limit 

d^ B->0 



[5^/.*] =,«.[.,„ (»)/«]=.. ,., 



The coordinates of E and -£"' are respectively /cos o', 
/cos/?, /cosy, and / cos (<ar + c^^'), /cos (/? + (J^), 
/ cos (y + ^?^)- Substituting in formula 

D = V[(:c" - x'r + (/' -yy + (z" - z'Yl 

and dividing by /, we have 

EE'/l = i/([cos {a + 6 a) - cos o']" 

+ [cos {(3 + d/?) - cos ftY + [cos (y + (J;/) - cos yY)- 

Dividing by ds, taking the limit as 6s b-» o, whence 
dipM-^o, substituting Sip for EE' /l, (2), we have (i) and 
§170 for the rate of curvature at M, 



limit r^"|_ 4// ^ cos <y y / ^cos /? y /^_cosj^y 

(3) 



(5 

^^ I 



ds p 
From § 88, 

cos a = dx/dsy cos /? = dy/dsy cos >^ = dz/ds. 



CURVATURE OF CURVES. 303 

Differentiating and substituting in (3), we have 

in which s is the independent variable. 

In order to obtain a more general expression for i/p, 
place (§115) 

d'^x _ dsd'^x — dxd'^s d'^y _ dsd'^y — dyd'^s 

~dl ~ ds' ' 'dF~ ds' ' ^^^'^ 

giving 

/{dsd'x - dxd'sf + {dsd'y - dyd'sY ' 

L= / -^-idsd'z-dzd'sY 

P V 'd? 

which may be written 

/ds\{d^xy -\- {d^yY + {d^zy\ 

1 ^ / -2dsd's{dxd'x+dyd'y+dzd''z)^-{d"'sY(dx''-Vdy''-^dz'' ) 

p V ^^' 

From § 88, ds" = dx' 4- dy' -\- dz\ 

Whence, dsd'^s = dxd'x + dydy-{- dzd^z. 

Therefore, 



p ~ ^ ds^ * ° ^5; 

which is a general expression for the rate of curvature at 
any point of any curve. 

If the curve is of single curvature, its plane may be taken 
as that of XV, z will be zero, and (5) reduces to (i) (§ 174). 



304 



DIFFERENTIAL CALCULUS. 



CHAPTER XVI. 



INVOLUTES AND EVOLUTES. 



178. Each point of any given curve as MM'M"y has, 
in general, a centre of curvature. 
The locus of the centre of 
curvature of any given curve is 
called its evolute. Thus, CC'C 
is the evolute of MM' M". 

The given curve ^^ J/ 'J/'' is 
called an involute of its evo- 
lute. 

179. Coordinates of the Centre 
of Curvature.— Let C be the 
centre, and CM = p the radius of curvature corresponding 
to My whose coordinates are x,y. Then 





OA = OP — AF, or x = x — psmil^\ 
AC = AB -^ BC, or 'y = y -\- p cos. tp , 



. . (i) 



^, and (§172) J = 



INVOLUTES AND E VOLUTES, 305 

which correspond with (2), § 172, since (§ 170) 

dx/ds = cos ip, dy/ds = sin ^, giving 

• . _ ds^dy _ dy _dy dx ^dy V /^V~l /^^-^ 
^^^"^"^ ~d^Js~dtp ~~dx'd^ ~^L W J/^^'^' 

180. Differentiating (i), § 179, with respect to s, we have 

dx/ds = dx/ds — p cos ipdip/ds — sin ipdp/ds. 



(i) 

dy/ds — dy/ds — p sin rpdip/ds + cos ^dp/ds. 
But (§§170, 172) 

^^y^j = i/p, cos ^ = dx/ds, sin ^ = ^/^^. 

Therefore, dx/ds = — sin ^dp/ds, 



dy/ds = cos il'dp/ds. 

Hence, by division, dy/dx = — oot tp. 

Represent the angle which a tangent to the evolute at 
yXf y) makes with X by ^'r, then, 
dy/dx = tan ^\ = — cot i/j = — dx/dy, or ^, = ^ + 7r/2. 

Therefore, the tangent to the evolute at (>x, 7) is normal 
to the involute at {x,y); or, in other words, the radius of 
curvature of any curve at any point is tangent to the evo- 
lute at the corresponding centre of curvature. An evolute 
must therefore be drawn tangent to all radii of curvature of 
the involute. An evolute is therefore the li'mi'f of the locus 
of points of intersection of adjacent normals to the involute, 
as the number of normals corresponding to any definite 
portion of the involute is increased without limit. 



3o6 



DIFFERENTIAL CALCULUS. 



It follows that a radius of curvature which is unlimited 
in length is an asymptote to the evolute. Hence, in general, 
the normal at a point of inflexion is an asymptote to the 
evolute. 

Also, when in the vicinity of any point a curve is sym- 
metrical with respect to the normal at the point, the corre- 
sponding point of the evolute is a. cusp. 

l8i. Squaring both members of (2), § 180, and adding 
each to each, we have 

{dx' + d folds'" = dp'/ds\ Hence, dp = ± Vdx'-\- d/- 
Let s represent the length of a varying portion of the 
evolute, then (§ 87) ds = ^ dx^ -\- dy^. Hence, dp = ds^ and 
(§ 74) p = -y ± ^, in which dJ is a constant. 
M" 

Let C'M'= p\ and C"if"= 
„ p", be the radii of curvature 
corresponding to M' and M'^ 
respectively. Measuring the evo- 
lute from C, and denoting the 
lengths of the arcs CC and CC 
by j-j and ^2 respectively, we have 




p' — s^-{- a; 



p" = s,-{-a. 
s, = arc C'C'\ 



(i) 



Hence, 

p"_p' = j, -^^ = arc C'C", . . . (2) 

that is, in general, ^Ae difference between any two radii of 
curvature of an involute is equal to the arc of the evolute be- 
tween the correspondifig centres of curvature. 

Exceptions exist when the arc of the evolute includes a 
singular point or is discontinuous. 



INVOLUTES AND EVO LUTES. 307 

Measuring the evolute from C, we have 

arc CC = p' — p, or p' = arc (TC -4- p; 
also, arc CC" = p" - p, or p" = arc CC" + p. 
Hence, (i), a =^ p. 

Similarly it may be shown that the constant a in equation 
(i) is, in general, equal to the radius of curvature which 
passes through the point of the evolute from which it is 
measured. 

It follows that as a right line rolls tangentially upon, or as 
a string is unwound from, any curve, each point describes 
an involute to the given curve as an evolute. Hence, 
while an involute has but one evolute, each evolute has an 
unlimited number of involutes. Any two involutes corre- 
sponding to the same evolute are separated by a constant 
distance measured along the normals, and are called parallel 
curves. 

182. In general, at cusps of the first species and at 
points of inflexion p changes sign. Hence, at such points 
we have p = o, or 00 , and the evolutes pass through these 
points or have infinite branches to which the corresponding 
normals are asymptotes. 




308 



DIFFERENTIAL CALCULUS. 




At a cusp of the 2d species p does not change its sign, 
and the corresponding point of the evolute will, in general, 
be a point of inflexion or a cusp of the 2d species. 




183. Radius of Curvature of an Evolute .—The angle 
M" ^^> between any two tangents 
as those at M' and J/", is equal 
to that between the correspond- 
ing radii of curvature to the in- 
volute, and since these radii of 
curvature are tangents to the 
evolute, the angle which they 
make with each other is equal 
to C"OC\ included between 
the corresponding radii of cur- 
vature of the evolute. 




INVOLUTES AND EVOLUTES. 309 

Let J", = CC\ Ss\ = C'C" ^ and pi = radius of curvature 
of evolute at C , 

Then(§x7o) p. = ^H^^r|]. 

But Ss^ corresponds to d^ = M'M" of the involute, 
and vanishes with it. Also (§181), Ss^ — f)" — p' = Sp. 

Hence p - ^'"^'^ T^l - ^ _ ^ f^^ 

in which s is the length of an arc of the involute measured 
up to M' and tp is the angle which the tangent at M' 
makes with a fixed right line. 

184. Equation of the Evolute — Let 

y=f{^) <0 

be the equation of any given plane curve. The coordinates 
of the centre of curvature for any point {x^ y) of the curve 
are (i), §172, 

^-[i+7V)VW//"W;, 

(2) 



y =j^+(i+/»)//"W. 

Expressions for /'(^) and /"(^) in terms of x andjj^, 
obtained by differentiating (i). substituted in (2) give x 
and y in terms of x and y. Combining these equations 
with (i), eliminating x and y, we have y = -F{x) for the 
evolute oiy =/(x). 

EXAMPLES. 

I. y^ = 2px. 

dy/dx = p/y, d'y/dx' = - ///. 

Substituting in (2), we have 



X =^ X 






310 



DIFFERENTIA L CA LCUL US. 




Combining with y" = 2px, and eliminating x and y 
we have 

for the evolute of the parabola, 

CCC" is the evolute of the parabola C OC. 

y =z O 
ives X =■ p = OC, 
x= 4p = OP 
gives 

y = ±pVS= ± PC" 

for points common to the pa 
rabola and its evolute. 

The arc CC"^ (3 4/3 - i)/. 
Transferring the origin to C, the axes remaining parallel, 
we have, denoting the new coordinates by x and j, 

^ = / + ^, y —y- 

Hence, we have ^ = ^x^/{2']p) for the evolute. The 
branch CC belongs to OC and CC to OC. 

Let r = FM' represent the focal distance of any point 
as M\ and let / = FY represent the perpendicular from 
the focus to the tangent TM'. 

Then FY' = FM' X FO, or P = pr/2. 

Hence, 2ldl = pdr/2, or dr/d/ — 4//^. 

From § 176, c = chord of curvature through F^ 
= 2/dr/d/. 
Hence, c = SP/p = Zpr/2p = 4r = 4FM' . 

That is, in a common parabola the chord of curvature 
through the focus is equal to four times the focal distance 
of the point of tangency. 



INVOLUTES AND E VOLUTES. 



311 



2. ay + '^V = ^'^'• 

dy/dx = - /?'x/ay, d'y/dx^ = - b'/ay. 

Substituting in (2) and reducing, we have 

^ = («^ - b')x'id; :. X = (a'x/[a' - d'])yK 

J = - {a^ - b^)yyb^; .'. y = - {b^/W " ^Y^^- 
Combining with a'y'^ + b'^x'^ = a^h^, we have 

{^xY"> + {b'yf^'' = (a' - b'f'^ 
for the evolute of the ellipse. 




CCC'C" is the evolute of the ellipse MM'X; x = o 
gives J = {a' - b')/b= OC" . 

When e" = {a'' — b'')/d' =1/2, we have a — 2b\ and 
OC" = b, in which case the vertices C" and C are on 
the curve. They are without (as in the figure) or within 
the ellipse according as e"^ is greater or less than 1/2. 

pleasuring the evolute from C, we have p = MC = b'^a, 
and p' = M'C' = a'/b. 

Hence, arc CC — a'^/b — b'^/a = (a — b>^)/ab. 
Axis CC" = 2a - 2b'' I a = 2{ar- — b")/a. 
Axis CC" = 2^ -h 2{a''/b - 2b) = 2{a' - b^b. 



312 



DIFFERENTIAL CALCULUS, 



Hence, the axes of the evolute are inversely as the cor- 
responding axes of the ellipse. 



3. ^ = r vers'-^ (jj'/r) — S/ 2ry — y^, 

dy/dx = {2rly - i)V2, d'^y/dx' = - r/y\ 

Substituting in (2), reducing and combining with the 
given equation, we have 

X =^ r vers"^ (— y/r) + ^ 



2ry — y 

for the evolute of the cycloid. 

Produce BA, making AO' = BA = 2r. 

B 



(a) 



o/_^ P 


r\ 


)^ 


p' 

X ^ 




/o' 



OM'0'0" is the form of an evolute of a cycloid. The 
branch 00' belongs to OB and O' O" to BO", 

Transferring the origin to (?', taking O' X' and O' A as 
the new coordinate axes, denoting the new coordinates by 
X and_y, we have for any point of the branch 00\ as M\ 

x = OA- O'P' = 7rr-x,y=:A0' -{■ P' M' = - 2r-\-y, 
which substituted in (a) give 



X =z 7Tr — r vers ' [(2^ — y)/r^ — y 2ry — J^ 
But 7ir — r vers"^ [(2r — y)/r~\ = r vers~^ (jA)* 



Hence, x — r vers ^ O'/') — V 2ry — y 



INVOLUTES AND EVOLUTES. 313 

is the equation of O'M'O referred to the new axes. It is 
of the same form and contains the same constants as the 
equation of the cycloid ; hence, the evolute of a cycloid is 
an equal cycloid. 

At O, p — o, and at B, p' = O'B — 4^. 
Hence, arc OM'O' ^ 2iYC OB = p' - p = ^r. 

Therefore, arc OBO" =^ Sr, that is, l/ie length of one 
branch of a cycloid is equal to four times that of the diameter 
of its generating circle. 

4. x^ -\- y^ = B\ X = y = o. 

5. ay - b'x^ = - a^b\ {axfl'' - (^j')^/^ --= (a' + b'fl^. 

6. x"^ = 4ay. x^ = 4{y — 2ay/2'ja. 

7. xV^ +//3 = aV\ (^+ J)^/' + {x -}f/^ = 2^2/3, 

8. 2xy = a\ (x + j')^/^ - (x - 'yf'' = 2^2/3^ 
185. Equation of Evolute in Polar Coordinates.— Let C 

be the centre of curvature corresponding to J/ of a curve 
referred to /* as a pole, and PX as a fixed right line. Draw 
/Tand PTx perpendicular to MT and MT^ respectively. 




T 

Let PM=r, PC^r., PT=p, and PTx=p,. 
Then r/ rz p^ -|- r' - 2rp cos PMC. 



3H DIFFERENTIAL CALCULUS. 

But r cos PMC — r sin PMT = p. 

Hence, r^^ = p" -\- r"" — 2pp. ..... (i) 

Also, ■ p,-' = TM' = r"" - p""; (2) 

and (i), § 176, p — rdr/dp (3) 

From the equation of the curve find p in terms of /', 
giving 

/-^W . (4) 

By eliminating r, p, and/, there results an equation be- 
tween r^ and p^ for the evolute. Thus, having r = a^, 
then / = cr^ in which c = 1/ i^i-{- log'«. (Ex. 3, § 150.) 

dp = cdr, dr/dp — i/c. 

Hence, f> = r/c, p^^ = r"" - c'r' = r\i - c"), 

and r^' = r'/c' + r' - 2r* = r\i -^ c')/c' =p,yc% 

or p^^ = c^r^ and /, = cr^^ 

for the evolute, which is a logarithmic spiral similar to the 
given curve. 

186. Having p in terms of ?/', equation (i), § 183, enables 
us to express p, in terms of tp^. Thus, 

r = ^a sin' (^A). 

Example 12, § 173. When tp = 3^/2, 

p = Sa sin (0/2)/^. 
Hence, p = 8^ sin (^/3)/3. 

(0, §183, 
p, = ^/p/V/^ = 8« cos (?/V3)/9 = 8^ sin (7r/2 + ^V3)/9- 
Let y^\ represent the angle which p makes with X, 



INVOLUTES AND E VOLUTES. 315 

Then ^ = ^, - 7t/2, and 11/2 + ^'/a = (/'^ + ;r)/3. 
Hence, Pj = 8^7 sin [(^^ + 7r)/3]/9. 

187. Equation of an Involute — Combine the equations 

~y = ^W, (i) 

dy/dx = — dx/dy^ (2) 

{x — x)^{y—y)dyldx-o, , ... (3) 

eliminating x and J^^, and there will result in general a differ- 
ential equation involving x and ji^. 

188. Involute of a Circle.— Let a = radius of circle, 
and let A be the initial position of generating point. Let 
TB = circum. AT h^ any position of the tangent rolling 
upon circum. AT. Take origin and pole at O. Let 




^ = angle AOT^ and let B = angle AOB, which radius 
vector r makes with X. Then tangent TB — aff^', and we 
have, for the rectangular coordinates of any point, as B, by 
projecting the two lines OT and TB upon X and F re- 
spectively, 

X — OB = a cos ^ + , , 

y=OQ=^a sin tp -' '" ^ ■ ' 



^^^ sin tpy ) 
arp cos ip. ) 



3l6 DIFFERENTIAL CALCULUS. 

In polar coordinates we have, from the right triangle 
TB, 



Vr' -a' = aip. 
But 6* = ^ - angle BOT = tp - sec"^ {r/a). 
Hence, aip = a6 -\- a sec'^ (V^)» 



and y /•' — a^ =aO -j^ a see"* (r/a) (2) 



ORDERS OF CONTACT OF CURVES. 



317 



CHAPTER XVII. 

ORDERS OF CONTACT OF CURVES AND OSCULATING 

LINES. 

189, Let y = f{x) and y = ^{x) be the equations of any 
two given lines, as BB and Z>Z>, which have in common a 




Y 


D M 




D 




/ 


B 




B 


\ 







t 
P 


P 


p" 





point M, whose abscissa is OP = a, and whose ordinate is 

PM^b=Ad) = <p(a). 

Increase a by an infinitesimal A = PP" ^ and we have for 
the corresponding ordinates, P" B and P" D^ designated by 
y andy respectively (§ 124), 

/ =Aa + h) =f{d) +/'{a)h +/"(a)h'/\2_+ ... (i) 
/' = (t>{a + h) = <t>(d) + 4>'{a)h + ct>"{a)h-'/\2 + ... (2) 
Subtracting (2) from (i), member from member, we have 



y -y 



DB=[f'(a)-4>'{ay\h 



+ [/"(«) -0"WF/|l+... (3) 
When /'(«) = 4>'(a), 
/ - /' = DB = [/"(«) - 4>"{aW/\2 

+ [/'"W - 0"'(«)F/|3|+ • • . (4) 



3l8 DIFFERENTIAL CALCULUS. 

the lines are tangent to each other, and are said to have a 
contact of at least the first order. 
When, also,/ "{a) = (p''{a), 

/ - /' ^nB= [/"' W - 0'" W]/^y|3 + . . . (5) 

the lines have a contact of at least tke secoiid order. 
V^htn,alsoJ''\a) = cp"\d), 

/ - /' = DB = [/-^^(a) - 0iv(^)]^y|4 + . _ (6) 

the lines have a contact of at least f/ie third order ; and, in 
general, the order of contact of any two lines having a point 
in common is denoted by the greatest number^ beginning 
with the first, of successive derivatives of their ordinates 
corresponding to the common point, which are, respectively, 
equal each to each. 

Denoting the order of contact of any two lines by n, we 
have also f^ia) — 0"(^), making with f{a) = (p(a), n -\- 1 
conditions, and giving 

From which we see that when {n + i) is odd, the sign 
of y — y" = DB will change when the sign of h changes, 
that is, if D is above B when h is negative and vanishing, 
it will be bl;low B immediately after h becomes positive. 
The lines will then intersect, as shown in Fig. i. When 
{n -\- i) is even, the sign of j' — y" — DB will not chanee 
with that of h, and the lines will not intersect. (See Fig. 2 ) 

Hence, when the order of contact is even, lines intersect at 
the common pointy a7td when it is odd, they do not. 

To illustrate, take the two equations 

/==4^, . . . (i) and (^ - 5)^ -{- (7 + 2)^ = 32. (2) 



OSCULATING LINES. 319 

Combining, we find that the point (i, 2) is common, 
(i) gives/'(i) = I, /"(i) - - 1/2, and f"\i) = 3/4. 
(2) gives 0'(i) = I, 0"(i) = - 1/2, and 0'"(i) = 3/8. 

Hence, the circle (2) has a contact of the second order 
with the parabola (i), and intersects it at the point (i, 2), 

Determine the common points, and order of contact at 
each, in the following pairs of lines : 



\y = x-\- I. 



, (i, 2) m common. 
Ans. ■{ ^ 

Contact of ist order. 



2. \ , Ans. (i, i) 2d order. 

ly = 3x — sx-\ri. 

{Ay =■ x"^ — 4. 
o , u , Ans. (o, — i) 3d order. 

r + ^ = 2)^ + 3- 

Two lines having at a common point a contact of the «th 
order with a third line, have a contact with each other of 
at least the ;^th order. 

190. Osculating Lines. — The line of any species of line, 
which at a given point of a given line has the highest pos- 
sible .order of contact, is called an oscidatrix or osciilatiiig 
line. Thus, the circle which at the given point has the 
highest possible order of contact is called an osculating circle. 
The parabola of highest contact is called an osculating pa- 
rabola. 

To determine the equation of an osculatrix at a given 
point of a given line, assume the general equation of the 
species of line in its reduced form. 

The problem then is ta determine such values for the 
arbitrary constants contained therein as will cause the re- 
quired line to have the highest possible order of contact. 



320 DIFFERENTIAL CALCULUS. 

Since the osculatrix must pass through the given point, 
substitute its coordinates in the general equation, giving 
one equation between the required quantities, and diminish- 
ing the number of arbitrary constants by unity. 

From the general equation of the species and the equa- 
tion of the given line determine expressions for the succes- 
sive derivatives of the ordinates to include those whose 
order is denoted by the number, less unity, of the constants 
in the reduced form of the general equation of the species. 

Substitute the coordinates of the given point in each, and 
place the results corresponding to derivatives of the same 
order equal to each other. The resulting equations with 
the one before obtained will equal in number the required 
quantities, which in general may be determined. Their 
values substituted in the general equation will give the re- 
quired osciilat7'ix. The order of contact will in general be 
denoted by the number, less unity, of arbitrary constants 
entering the general equation, but in exceptional cases it 
may be higher. 

EXAMPLES. 

I. Find the equation of the osculating right line to the 
parabola y = 9^ at the point (i, 3). 

\x\ y z=z ax -\- b substitute the coordinates (i, 3), giving 

Z = a-\-b (i) 

From,)' ■= ax -\- b y^Q find 0'(^) = a. 
Fromy = 9^ we find/'(jc) = 9/27. 

Substituting the coordinates (i, 3) in each, and placing 
the results equal to each other, we have a = 9/6 = 3/2, 
which in (i) gives b = 3/2. Hence, ^ = 3^/2 + 3/2 is the 



OSCULATING LINES. 321 

equation of the required line, which is tangent to the 
parabola. 

Since the general equation of a right line contains but 
two arbitrary constants, it cannot, in general, have a con- 
tact of an order higher than the first with a plane curve. 

An exception exists at a point of inflexion where, in 
general, for both the curve and the right line, we have, de- 
noting the abscissa of the point by ^, f'ia) = o. The 
contact is, therefore, at least of the second order. 

At a point of inflexion the direction of curvature changes, 
and y' — y" = DB (Fig. i, § 189) changes sign with h. 
Equation (7), § 189, shows that this occurs only when n-\- \ 
is odd. Hence, the order of contact is even and the tangent 
intersects the curve. 

2. Find the equation of the osculating circle to the parab- 
olay = AfX at the point (i, 2). 

(x — ay ~{- (y — by = i?" is the general equation of the 
circle. Substituting the coordinates (i, 2), we have 

(,-ay + (2-dy = j^' (1) 

Differentiating the general equation of the circle, we find 
0'(a:) = -{x- a)/y - b, and 0"(^) = - R'/^y - by. 
From y = 4X we obtain 

/'{x) = 2/y, and /"W - - 4//. 

Substituting the coordinates (i, 2) in each, and placing 
the results corresponding to derivatives of the same order 
equal to each other, we have 

- (i - a)/{2 - ^) = I and - i?V(2 - by = -4/2', 
which with (i) give 

^ = 5j b ■= — 2, and i?' = 32. 

Hence, {x — ^y -\- {y -\- 2)^ = 32 is the required equation. 



322 DIFFERENTIAL CALCULUS. 

191. Osculating Circle at any point (^',/) of any plane 
curve whose equation isj^ =/(-^0. 

Substituting {x\y') in {x — a)' + {y— bf = R\ we have 

{x'-aY-\-{y-by = R'^ (.,) 

From {x — af -f (y — by = R"" we obtain 

n-) = -J^y and 0"(x) = -[i + (i^]']/(;,-^); 

and from y = f{x) we derive expressions for f'{x) and 
f"{x). 

Substituting (x', y) in each, and placing the results cor- 
responding to the derivatives of the same order equal to 
each other, we have 

-{x'-a)/{y'-b)=f{x'), 

and -[r+fWYy(y'-l>)^/"(x'), 

which with {a) give, omitting the primes, 



ji=[i+f(x)ryf"{x), (,) 

g := X - [I +f'{x)'y{x)/f"(x), . . . (2) 

^=7+[i+/'wV/"W (3) 

Comparing ihese with (i) and (2), § 172, we see that t/ie 
oscillatory cu'cle at any faint of a plane curve is the circle of 

curvature, 

EXAMPLES. 

1. Find the equation and radius of the osculating circle 
to the curve 4(7 -}- i) = ■^^^ ^.t (o, — i). 

Ans. y -f ^^ = 2j^ -[- 3 ; radius = 2. 

2. Find the radius of the osculating circle to the parabola 

y = QJJC, at (3, i/27). Ans. 16.04. 



OSCULA 7ING LINES. 323 

3. Find the equation and radius of the osculating circle 
to the parabola y'^ = i6jk: at (1, 4). 

Ans. {x — 11)^ + (>■ + i)^ = 125 ; radius = 5 4/5. 

192. In general, an osculating circle has a contact of the 
second order with any plane curve, but § 172 shows that at 
a point where the curvature of a curve is a maximum or a 
minimum, the circle of curvature, and therefore the oscu- 
lating circle, does not intersect the curve. The order of 
contact is therefore odd, and of a degree higher than the 
second. That the contact in such cases is at least of the 
third order may be shown as follows: 

From (i), § 172, and (i), § 191, we have 

^ = P=[i+7>)TV/"(^); 

and in order that p may be a maximum or a minimum, 



^ = ogives 3/ W y W - / WLi 


+ / i 


whence /'"(x) = 3/"{x)'/'{x)/[, + /'(x)']. 


From § 114 we have for a circle 


dy Id'yVdv /T , ld}^~ 
dx' - ^\dxV dxl L' "^ \dx) „ 


• 



Hence, when the radius of curvature is a maximum or a 
minimum^ the circle of curvature has a contact with the curve 
of at least the third order. 

It follows that the order of contact of an osculating circle 
at a vertex of a conic is odd, higher than the second, and 
the circle does not intersect the conic. 



324 



DIFFERENTIAL CALCULUS, 



CHAPTER XVIII. 



ENVELOPES. 
193. In 

u =f{x,y, a) = (i) 

let a be an arbitrary constant. By giving all possible values 
to ^, (i) will represent a series of lines, all of the same kind 
or family, unlimited in number, and, in general, intersecting 
each other in order, a is then called a variable parameter. 




To illustrate, let 

u = (x — ay -fy — 9 = 0. 

By giving different values to «, the equation may repre- 
sent a series of circles having the same radius, their centres 
on X, and intersecting each other in order in points as 
m^ m\ etc. 

In general, any value of a in (i) corresponds to a deter- 
minate particular line, and ^ + y^ to another line of the 
same kind having for its equation 

u' :=^f{x,y,a^h) = o (2) 



ENVELOPES. 325 

This second line, which may be regarded as a second 
state of the first, will, in general, as h vanishes, ultimately 
intersect the first, in points m^ m\ etc., the coordinates of 
which will satisfy both (i) and (2). If (i) and (2) be com- 
bined and a eliminated, the resulting equation will be that 
of the locus of the points m, m\ etc., of intersection of all 
of the series of lines represented by (i), each with its second 
state. If in this resulting equation h be made equal to zero, 
we will obtain the equation of the limit of the above locus, 
and this /////// is called an envelope of the series of lines. In 
the case of the circles, the right line MM' M" is the limit 
of the locus mni'ni" , and is an envelope of the circles. 

In combining (i) and (2) so as to eliminate a, compli- 
cated expressions frequently arise which may sometimes be 
avoided by the following method of Calculus. 

Since the coordinates of the points m, m\ etc., common 
to each of the lines of the series represented by (i), and its 
second state in order, satisfy both (i) and (2), they will 
satisfy the equation 

W - u)lh = [fix, y,a + h) - /{x, y, a)]//i = o. (3) 

As /i vanishes, the points m, ;//, etc., approach limiting 
positions M, M\ etc., and the coordinates of the points 
M, M\ etc., will satisfy both (i) and the equation 

'du/'da — 9/(^, y, a) /da = o, ... (4) 

which (3) approaches as /i vanishes. 

If, therefore, (i) and (4) be combined so as to eliminate 
a, the resulting equation will be that of the loci^s of 
the limiti?ig positions of the points of intersection of the 
series of lines represented by (i), each with its second 
state. This locus is the same as the limit of the locus of 



326 DIFFERENTIAL CALCULUS. 

the points of intersection, etc., before obtained, and is, 
therefore, an envelope of the series. 

Hence, an envelope of any series of lines determined by 
giving all possible values to a variable parameter in an equa- 
tion involving two variables only may be defined as the 
limit of the locus^ or the locus of the limiting positions of 
points of intersection of the series of lines, each with its 
second state, under the law that the difference in position 
between each second state and its primitive vanishes. 

To obtain the equation of an envelope of a series of lines 
given by an equation with a variable parameter, we have 
the following rule: 

Combine the given equation with its diffei-ential equation 
taken with respect to the variable parameter^ and eliminate the 
parameter. 

From (i), regarding a as constant, we obtain 

du/dx = 'du/'dx -\- {'dti/dy){dy/dx) = o 

for the differential equation of each of the lilies of the series. 
From (i) and (4), ^= <p{x), which substituted in (i) 
gives the equation of an envelope. Hence, differentiating 
(i) regarding a ■=■ <p{x), we have 

du _ c)u 'du dy c)u da 

dx 'dx dy dx 'da dx * 

or^ since du/da = o, 

du/dx = du/dx + {du/dy)(dy/dx) ■= o, 

for the differential equation of an envelope. Each line of 
the series, and an envelope, have, therefore, the same differ- 
ential equation, and dy/dx at any point common to any line 
of the series, and an envelope will be the same for botli. 
An envelope is therefore tangent to all of the lines of the series. 



ENVELOPES. 327 

EXAMPLES. 

1. Find the equation of an envelope of the series of circles 
given by the equation 

u—i^x — aY +/ — 9 — o, . . . . (i) 

when ^ is a variable parameter. 

du/da = — 2{x — a) = o. , . . . (2) 

Combining ^i) and (2) so as to eliminates, we have for 
the required envelope 

y= ±3, x= 0/0. 

Hence, the right lines MM'M" and MM^M^^ (see figure, 
§ 193) are the envelopes. 

2. Find the envelope of the curves given by 

_y = ^ tan 6 — x^ /{\h cos^ 0), as varies. 

du/d^ =^ jc/cos^ — x^ sin 6/2/1 cos^ = o. 

Hence, tan 6 =2/i/x, i -\- tsui^ 6 = {x'^-\- 4h'^)/x'^= sec^ ^, 

and cos' = x'/{x'-{- 4/1'), 4/1 cos' 6 = ^hx^/ix" + 4//'). 
Substituting in given equation, we find y =^ h — x'^/ij^Ji) for 
the required envelope. 

3. Find the envelope of a right line of a given length c 
which moves with its ends on the coordinate axes. 

Let B equal the angle which the right line makes with 
X. Then intercepts are, respectively, c cos and c sin By 
giving 

u =^ X sec -\- y cosec 6 — c =■ o . . . (i) 

for the line, in which is the variable parameter. 

'du/'dd = X sec tan 6 — y cosec 6 cot 6^ = 0.. (2) 

Combining (i) and (2), we find 

X ^=^ c cos^ 6^, y =^ c sin^ 0\ 



328 DIFFERENTIAL CALCULUS. 

and eliminating ^, we have for the required envelope 

Otherwise, let a and b represent the intercepts, respec- 
tively, giving 

x/a+y/b=i . . . (3) and a' -^ b'= c\ . . (4) 
Regarding b as the variable parameter, we may by means 




of (4) eliminate a from (3), and proceed as before ; or, since 
tf is a function of by we have 

xda/a' + ydb/b"" = o (s) 

ada -{- bdb = o (6) 

Combining (3), (4), (5), (6), eliminating da/db^ a and by 
we have 

{a^ + b'')y = b'; .'. b= (^»V3 , and ^ = {c^x^. s , 

Substituting these expressions in (3), we have for the 

envelope 

^2/3 _|_ ^2/3 — ^2/3 ^ 

In a similar manner, when there are n parameters and 
n ~ I equations of condition between them, we may differ- 



ENVELOPES. 329 

entiate the n given equations, regarding n—\ of the param- 
eters as functions of the variable parameter. Then, by 
combining the n differential equations with the given equa- 
tion of the series, the parameters may be eliminated and 
the envelope determined. 

4. Find the envelope of a series of concentric and co- 
axal ellipses having the same area. 

The given equations are 

«V + ^x" = a^b\ and ab - c\ 
xy = c^l2 is the envelope. 

5. Find the envelope of a right line moving so that its 
perpendicular distance from the origin remains constant. 

u = /(jc, y, a) ^^ X cos a -\- y sin « — / = o, 

'du/'da = — :r sin «f -j-jj; cos a = o, 

and x^ +y — p'^ is the envelope. 

6. Find the envelope of the hypothenuse of a right tri- 
angle moving so that the area of the triangle remains con- 
stant. 

Let a = constant area, b = base = variable parameter, 
and c = altitude = 2a/b. Then 2a/b'' = tan of angle 
hypothenuse makes with base, and taking the coordinate 
axes to coincide with the legs, we have 

y = 2ax/lP' + 2a/ b for the hypothenuse, 

and xy = a/ 2 for the required envelope. 

7. Find the envelope of a right line moving so that the 
sum of its intercepts on the coordinate axes is constant. 

Let a and b represent the intercepts upon X and Y 
respectively, and let c ^=- a -\- b\ then the equation of the 
line is 

•^A + j/(^ - ^) = I, 



330 DIFFERENTIAL CALChLUS. 

and a is the variable parameter. 

Ans. x^ -\-y^ — 2xy — 2cx — 2cy -\- c"^ = o 
or Vx -\- \^y = ^ c. 

8. Find the envelope of a right line moving so that the 
product of its distances from two fixed points is constant. 

Take X to coincide with right line joining the two 
points, and the origin to be at its middle point. Let (^, o) 
and (— «, o) be the two fixed points, and 

X cos a -\- y <sAXi a ^= p (i) 

the equation of the line. 

The distances from the fixed points to the line are, re- 
spectively, 

a cos a — p and — a cos a — p. 

Hence, p^ — a^ cos^ a = ^r^ = constant. ... (2) 

From (i) and (2), regarding a as the variable parameter, 
we find for the envelope 

9. Find the envelope of the right lines whose equation is 

y-f^2b{x-x')/{l. -b'^l . . . . (l) 

wheny is the variable parameter, and we have 
y'"^ = 2px' and /^ = — y' /p. 
Eliminating b and x\ (i) becomes 

JK^" -/>+// — 2//^ = o. ... (2) 
Hence, 'du/'dy' = 2yy' -\- p^ — 2px = o, 

and / = (2px — p')/2y (3) 

Conjbining (3) and (2), eliminating y, we have 

X^ ±V^'-\-p/2 



ENVELOPES. 331 

for the envelope, which is a point at the focus of the 
parabola, and is the Caustic of rays of light reflected from 
the concave side of a parabola, the incident rays being 
parallel to the axis which coincides with X. 

10. Find the envelope of the right lines whose equation is 

y-y'=.^i{x-x')/{l-b^),. . . . (l) 

when x^ is the variable parameter, and we have 

y^-|.^'^ = ^2 and d=y/x\ 

Eliminating d, (i) becomes 

:^-x'={//x'-x'//){/-y)/2. . . (2) 

Differentiating and reducing, we obtain 

du/dx'= v' - a'/y/' = o, or / = a'/y/'. . (3) 

Substituting, in (2), y'x^'^/a'^ for jv' — y, we have, after 
combination with (3) and reduction, 

x' = 2ay'x/{a' ' + 2yy'}. 
Squaring the expressions for y' and x', and substituting 
in y'^ -f~ "^'^ — ^^} we have for the envelope 

which is the equation of the Caustic of rays of light re- 
flected from a circle, the incident rays being parallel to X. 

11. Find the envelope of the polar line to the ellipse 
9/ + 4-^^ = 36 as the pole moves along the right line 
y ^^ 2x-\- \. 

Let x" and y" be the coordinates of the pole, giving 
y" = 2^' + I. 

Then ()yy" -\- /^xx" = 36 is the equation of the polar line 
Substituting 2x'' -\- 1 for 7", we have 

u = iSyx'^ + 9V + 4xx'^ — 36 = o. 



332 DIFFERENTIAL CALCULUS. 

in which x*' is the variable parameter. 

Combining the last two equations, we have ox^' -[- 9J = 2>^i 
or_>' = 4, .%• = — i8 for the required envelope. 

"nes. prrre^'/rs. Envelopes. 

\2. y = ax -^ bja. a. y^ ^ ^dx. 

13 ax — y = x'^{i -\- a^)/2p. a, x^ -{- 2py = p^. 

14. x^a +y''/ia - k)=i. a. (^x ± \/hf -f / = o. 

15. x"^ -\- y^ =: r^ . r. 

16 y^ = ax — a"^. a. y=± x/z. 

17. X cos 30+>' sin 30 

= a(cos 20)^/^ 0. (^'+/)" =«^^''-y). 

^ ■ U^ -f (;« - ^)2 = r\ \ n = f{m). + 2da-'x = a*. 

ig. ry^ = a'^x — a^. a. y^ = 4x^/2^^ 

Ux-ar-^{y-dy = rK <a. 



;r^ +/ = (^ ± r)\ 
21. y = 2ax-\-a*. a. \t>y^ -\^ 2-^x^ — O. 

22 y'^A^iyX — a^ — 2pa = O, a. y"^ =p{p + 2x). 

X =0. 

y=± cx/(i - c^). 



256/^ -f- 27X* = o. 



23- 


/ = 2px. 


P- 


24. 


\ r — ca. 


a. 


25- 


y"^ =1 ax — a'^. 


a. 


26. 


y = ax -\- a*. 


m. 




a"" cos d^ sin ^» 


6. 


27- 


X y a 


28. 


y — mx -\- i^a^m^ + dK 


ni. 


29. 


i {x/af + {ylb)- = I. 
1 a-^b = c. 


a. 



ENVELOPES. 333 

194. The envelope of the normals of any given curve is 
its evolute. 

This follows from the definitions of envelopes and evo- 
lutes ; otherwise, the equation of a normal to a curve 
>'=/(^), at (^',/), is (§148) 

x-x'-\-{y-y)f{x') = o, . . . (i) 
in which y' = f{x'), and x' may be taken as the variable 
parameter. Differentiating with respect to x', we have 



-I-/V) +(^-y)/'V) = o. . . (2) 

Combining (i) and (2), we find 



x = x'-ii +f{x'Y]f{x'/f"{x'), , 

(3) 



y=y + b+f(xyyf"(x'), 

for the limiting position of the intersection of the normal at 
{x\y') and its second state ; which is therefore the point of 
tangency of the envelope to the normal at {x',y'). Com- 
paring (3) with (i), § 172, we see that this point is the cor- 
responding centre of curvature of the given curve. Hence, 
the envelope of the normals is the evolute of the curve. 

Combining (3) and y' =f(x'), x' may be eliminated and 
the equation of the envelope obtained. 

EXAMPLES. 

I. Find the envelope of the normals to the parabola 
Here / ^/(^') = 2^VVV2, 

f{x') = a'/yx''/\ f"{x') = - ayy{2x'y% 

Substituting in (3) and eliminating y, we have 

, (i + a/x')(ay'/x'y') _ 

•^ - ^ - ayy{2xy/'' - 3^ -t- 2^, 
y=y+ ltyv.% --^^'''Vay% 



334 DIFFERENTIAL CALCULUS, 

from which, eliminating x\ we obtain for the required en- 
velope ay'^ = 4(x — 2ay/2']. (See Example i, § 184.) 

2. Find the envelope of the normals to an ellipse by the 
above method. 

Otherwise, the equation of the normal to an ellipse at a 
point whose eccentric angle is denoted by 6^ is 

u^=^ ax sec 6 — by cosec — a^ -\- F =■ o. 

Regarding 6 as the variable parameter, 

du/dG = ax sec B tan d -\- by cosec 6 cot 6 = 0. 

Combining and eliminating 6, we have for the required 

envelope (ax)y^ + {byf/^ = {a' - b'f/K 



CURVE TRACING, 



335 



CHAPTER XIX. 



CURVE TRACING. 



I95* The foregoing principles, with those from Analytic 
Geometry, enable r.s, in general, to trace curves from their 
equations with great accuracy. 

RECTANGULAR COORDINATES. 

No fixed rule or directions ap]:)ly in all cases, but, in gen- 
eral, it is desirable to determine — 

i". Symmetry with respect to the coordinate 






Limiting coordinates and asymptotes paral- 
lel to them. 
Points on the coordinate axes. 
Terminating points. 



b/3 tn ^ C 

crts I- u 



1> 13 o l> 

llll 



5°. Direction of curve at points on the coordi- 
nate axes. 
6°. Asymptotes oblique to coordinate axes. 
7°. Multiple points. 

8°. Character of cusps. 
9°. Maximum and minimum ordinates. 
io°. Direction of curvature and points of \\v 
flexion. 

EXAMPLES. 

Each value of x from — oo to -|- ^ gives a real value 



33^ 



DIFFERENTIAL CALCULUS. 



for y. The curve is therefore unlimited in both directions 
with X, and is limited in both directions of Y. 

As :x: B-> ± 00, J ^H^ ± o. Hence, X is an asymptote in 
both directions. 



X =^ o gives ji^ 



o, and y=^ o gives a; = o, or ± oo . 
Y 





Hence, (o, o) is the only point at which the curve cuts 
the coordinate axes. 

f{x) = a\a' - x')/{a' + x')\ 

f\6) = I. Hence, the direction of the curve at the ori- 
gin makes an angle of 45° with X. 

f'(x) = o gives x-^ ± a and ± co . 

f'{x) = 2a'x{x' - z^")/{a' + xy, 

f'\a) is negative; /. y = a/ 2 is a maximum. 

/"( — a) is positive; .'. j^ = — a/ 2 is a minimum. 

f"{x) = o gives :r = o and ± a V^. 

f"{x) is negative for values of x from — 00 to — ^ 1^3, 
and the curve is concave downward. 

For values of x from — a V^to o f"(x) is positive, and 
the concave side is above. 

As x varies from o to a V$, /"(•^) is again negative, and 
the concavity is downward. 

Values of x from « 1^3 to + 00 make f"{pc) positive, and 
the curve is concave upward. 



CURVE TRACING. 



337 



It follows that 

( - a 1/3, - « ^3/4), (o. o), and {a 4/3, ^ 1/3/4) 
are points of inflexion. 

2. ;t:' — 2x'y — 2^' = 8>'; .'. y — x'ipc — 2)/2(^' + 4). 

^ = — 00 = J. :v: =: o — J'. jc = 00 = J. 

The curve is unlimited in both directions along X and Y, 
y ^=. o gives jc = o and 2. Hence, the curve cuts X at 

the points (o, o) and (2, o). 
f\x) = ^(;(;^ + 12;^ — i6)/2(^* + 4)^- 
/'(o) = o. Hence, at the origin X is a tangent. 
/'(2) = 1/4. Therefore, at the point (2, o) the curve 

makes tan"^ (1/4) with X. 
f\x) = o gives .T = o and 1.19 nearly. 




Expanding the expression for j', we have 

y — x/ 2 — I + (4 — 2x)/{x^ -\- 4), in which, as x b-^ 00 , 

y :m-^ {x/2 — i). 
Hence, _>' — x/2 — i is an asymptote. 

f\x) := - 4{x' - ex' - I2X + S)/{x' + 4)'. 

/"(o) is negative ; hence, j^ = o is a maximum. 

/"(1.19) = o is positive; hence, j == ~ o.ii is a 
minimum. 

f''(x) = o gives ;t:— — 2, 4—21/3 = 0.54, and 
4 + 2i^= 7'5- 



338 



D IFFEREN TIA L CALCUL US. 



Vallies of X from — oo to — 2 make f"{x) positive, and 
the corresponding part of the curve is concave upward. 
As X varies from — 2 to 0.54, f"{x) is negative, and the 
concavity is downward. When 0.54 < x < 7.5, f''{x) is 
positive, and the concave side is above the curve; but 
X > 7.5 makes f"{x) negative, and the concavity is down- 
ward. It follows that (— 2, — i), (0.54, — 0.05), and 
(7.5, 2.6) are points of inflexion. 

2,. y = a^x/{x — a)'. 

f\x) = - a\a + x)/{x - a)\ 

f\x) = 2a'{x + 2a)/{x - ay. 

The curve is unlimited in both directions cf X, and in 
the positive direction of Y. 

j; = o gives X ^= o and ± co . 

X is an asymptote in both directions, and since j3»-> 00 
as X -w^ a, jc = « is an asymptote. 

/'(o) = I. Hence, at the origin the curve makes 
tan"^ I with X. 

f'{x) = o gives X = — a, and /"(— a) is positive. 
Hence, y = — a/4 is a minimum. 




To the left of the point of inflexion (— 2a, — 2a/g) 



CURVE TRACING, 



339 




the concave side is below, and to the right it is above, the 
curve. 

4. y^ = 2ax^ — x^. .*._)/ = x^/'^{2a — xY^^, 

f{x) ={4ax-sx')/sf. 

f'{x) = -Saygx''/%2a-xy/^ 

The curve is unlimited in both 
directions along X and F. It cuts 
X at (o, o) and (2a, o). V is 
tangent to both branches at the 
origin, which is a cusp of the first species; and the tangent 
at {2a, o) is perpendicular to A', y — — x -\- 2^/3 is the 
equation of an asymptote in both directions, y = a ^32/3, 
corresponding to jv = 4^/3, is a maximum. (2^, o) is a 
point of inflexion to the left of which the curve is concave 
downward, and to the right of which it is concave upward. 

f(x) ={3- xYis - 5^)/i6. 
V"W = (3 - ^nS^ - 6)/4. 
Y 




As X B-^ ± 00 , j; B-> ± 00 . X — o = y. The curve is 
unlimited in the directions of X and V. {o, o) and (3, o) 
are points on X. /'(o) — 5.06, and/'(3) = o. /'(x) = o 
gives X = 3/5 and 3. fis/s) is negative; hence, y = 1.24 
is a maximum. /'(^) changes sign in passing through 



340 DIFFERENTIAL CALCULUS. 

/'(t,) = o (§ 135), and 7 = o is a minimum. (1.2, 0.79) is a 
point of inflexion to the left of which the curve is concave 
downward, and to the right of which the concavity is up- 
ward. 

6. / = x^ + x\ 



f'\x) = ± {isx' + 24x-{-S)/4{x+iy/^ 
The curve is symmetrical with respect to X, 

Y 




Values of x <. — 1 give imaginary expressions, for j. 
X — — I gives y = ± o. x > — i gives two values for/ 
equal with opposite signs. As x :^-^ 00 , y b-> ± co . The 
curve is, therefore, limited in the direction of JC negative 
by the ordinate corresponding to :r = — i, and is unlimited 
in the other directions along X and V. 

(— I, ± o) and (o, ± o) are points on X. 

/'(— i) = ± 00, and /'(o) = ± o. Hence, at (—1 = 
± o) the tangent is parallel to V, and X is tangent to both 
branches at the origin, which is a multiple point of tangency. 
(Example 6, page 281.) 

f'{x) — o gives X — o or — 4/5. /"(o) is positive for 
the upper and negative for the lower branch. Hence, the 
zero ordinates at the origin are, respectively, a minimum 
and a maximum. /"(— 4/5) is negative for the upper and 



CURVE TRACING, 



341 



positive for the lower branch. Hence, the corresponding 
ordinates are, respectively, a maximum and a minimum. 

f"{x) — o gives x = (— 12 ± 1/24)715. Points of both 
branches corresponding to the upper sign are points of in- 
flexion, and the direction of curvature is as indicated in 
the figure. 



/'W = ± 



a^ -\- 2c^x'^ — x^ 




The curve is symmetrical with 
respect to X and Y. It is limited 
in both directions of X by the 
asymptotes Jt: = ± a, and is un- 
limited in both directions along 
F. 

/'(o) = ± I, and /'(± ^) = ± «). 

Both branches pass through the origin, one inclined at 
an angle of 45°, and the other at an angle of 135°, with X. 
f\x) is an increasing function for the branches above A, 
which are, therefore, concave upward, and a decreasing 
function for those below X, which are concave downward. 
8. J = i^V^ 

f\x) = i/x'e^/\ 
f'\x) = (i - 2x)lx'ey\ 



As X ^-> =F 00 , jv :^-> I. K% — X m^ o, y b-> 00 . As 
-f- X 'Wf-^ o, y M-> o. 

The curve is limited by AT in the direction of F negative, 
and is discontinuous at the origin which is a terminating 



342 



D IFFERENTIA L CALCUL US. 



point for the right-hand branch. As — :r -m-^ o, f'{x) -b-^ oo ; 
and as + ^ B-> o, /'(^t) ;^-» o. Hence, X is a tangent at 
the origin. F is an asymptote to the left-hand branch, and 
J = I is an asymptote to both branches. As x varies con- 




tinuously, /'(jk:) does not change sign, and there are no max- 
imum or minimum ordinates. Corresponding to :r = 1/2, 
there is a point of inflexion, to the left of which the curve 
is concave upward, and to the right of which the curvature 
is downward. 
9, The Logarithmic Curve. 

X = ey, .*. y = log X. 

fix) = i/x. f'{x) = - ^|x\ 



As jc^-^o, y-^^ — 00. x—1, 
jj; = o. As Jts^^oo , jj;;^-»oo . The 
curve is limited in the direction 
of X negative by F, which is an 
asymptote, and is unlimited in 
the other directions along X and 
F. /(i) = I. Hence, at (i, o) 
the curve makes tan"^ i with X. 
f\x) is negative for all points of the curve, and the con- 




CURVE TRACING. 



343 



cave side is below. RS, the sublangent on F, is (§ 149) 
xdy/dx = I = 6>.-^. 
10. y = ax" -|- bx^. 




f(x) = ± (^? + ^bx/2)/ Va + bx. 
f'{x) = ± {4ab + 3^'^)/4(a + ^^Y''- 

The curve is symmetrical with re° q 
spect to X. 

As :r^-»oo , j:^-> ± 00 . x = o = J. 
^ = — ^/<^, y = ± o. X < — a/b, 
y is imaginary. 

The curve is limited in the direction of X negative by 
the ordinate corresponding to x = — a/b, and is unlimited 
in the ether directions along A" and Y. 

f{~ a/b) = ± CO . f{o) = ± \/a. The origin is, there- 
fore, a double multiple point. 

f'{x) — o gives X = — 2a/;^b, for which j^' = ± 2a V^^/gb 
are maximum and minimum ordinates. 

For ^ > — a/b the first value of f (x) is positive, and 
the second negative. The branch BEOC is, therefore, 
concave upward, and the other is concave downward. 

POLAR coordinates: examples. 



I, r = rt; sin 2^. 

dr/cW = 2a cos 26. 
As varies from o to 7r/4, r 
X changes from o to a, and as 6 varies 
from 7r/4 to 7r/2, ; changes from a 
to o; completing a loop in the first 
angle. As f^ varies from 7r/2 to tt, 
is negative, and changes from o, through — a^ to o form- 




344 



D IFFEREN TIA L CALC UL US. 



ing a loop in the fourth angle. As varies from n to s^r/?, 
r is positive and changes from o, through a^ to o, forming 
a loop in the third angle. As ^ varies from 37r/2 to 2;r, r 
is negative and changes from o, through — a^ to o, forming 
a loop in the second angle. 

As 6^ passes through 7r/4, dr/dd changes from + to — . 
Hence, r = (^ is a maximum. As d passes through 37r/4, 
dr/dd changes from — to -|-> ^^d r = — « is a minimum. 
As 6* passes through 57r/4, dr/dd changes from + to — , 
and r = « is a maximum. As B passes through 77r/4, dr/dd 
changes from — to +, and z' = — ^ is a minimum. 

^ 2. r = a tan 6. 

dr/dd = a/cos' d. 

r is always an increasing func- 
tion of d. 

Values of d from o to 7r/2 give 
the branch FM\ As d varies 
from 71/2 to rr^ r is negative and 
increases from — 00 to o, giving the 
branch in the fourth angle. Values of d from TT^to Z'^/2 
determine the branch in the third angle, and the branch in 
the second angle is due to negative values of r correspond-, 
ing to values of d from Z'^/2 to 2 7r. 

The subtangent = r''dd/dr — a sin'' d. 

— 7t/2 or 37r/2 gives r = 00 , and the subtangent = a. 

Hence (§ 157), ^ cos d = ± a are asymptotes. 

3. The Spiral of Archimedes, r = ad. 

Estimating from the pole J^, where r = o = d, r in- 
creases directly with d. Denoting the value of r after one 
revolution by ~ 




CURVE 7' RACING, 
r, = PA ~ 27ta, 



345 




we have a = rj27t, 
and r = r^6/27t. 

Hence, 
FO = ry4, ^(9' == rj2, 
PO" = r,3/4, etc. 

dr = r^dB/27C ; 
.'. dQIdr — 27t/r^, 
Subtangent — r'^dO/dr 
= rj)'^/27t, (§ 150.) 

Hence, subtangents are to 
each other as the squares of 
the corresponding radii. 6 — m2n gives subt = m^27ir^. 

Subnormal — dr/dO = rj2n, 

4. The Parabolic Spiral, r'' = d'd. 

r^ = aVzTt', .*. a = rj^27t^ and r" = r^B/27t. 

dr/d6 = (f/2r = subnormal. 
Subtangent = 2r' /c^. 

This spiral may be con- 
structed by first construct- 
ing the parabola y^ = x^ and 
the circle CB with centre at 
P and radius = «^ Then 
lay off from C the arc CB equal to an assumed abscissa of 
the parabola, and upon the radius PB lay off from P PO 
equal to the corresponding ordinate. O will be a point of 
the curve, since r = PO =: y = Vx — Vd^O. 




34^ DIFFERENTIAL CALCULUS, 

5. The Hyperbolic Spiral, r — a/6. 

r^ = a/27t; .'. a = 2nr^ and r = 27trJ6, 
B C 



V 


0' 


^ 


r,/^ 


M 




A 


?f 


\ 






A 


Vj 


J 











0'" 

At the pole r = o and ^ = 00 . r varies inversely with 
e. Hence, FO' = 4r, , ^6"' = 2r, , /'C)"' = 4^73, etc. 
^/-/^//^ = — a/d''-^ subtangent PT = —a. 

6 := o gives r =: 00 , and the subtangent PB := — a. 
Hence (§ 157), PC, parallel to PA, is an asymptote. 

From § 150, tan PAfT= rdd/dr = — 6^, which leads to 
a construction of the curve by points. Thus, with /^ as a 
centre and radius = a, describe a circle. Draw any radius 




vector as PM, and the corresponding subtangent PT, 
Lay off PE — arc HN, and draw ElSf. TM drawn par- 
allel to EIV \N\\\ determine a point M' oi the curve; for 
tan PMT = tan PJVE = PE/ci = HNJa ^ 6. 



CURVE TRACING. 347 

6. The Logarithmic Spiral, r — a^. 

dO/dr = Mjr, whence (§ 150) tan/'J/r = tan =: M^; 
and is constant. 

Also, sin = rdd/ds. 

Hence, ^ = r sin = r^dO/ds = cr^ 

in which c represents the sine of the constant angle made 
by the radius ^^ector with the curve. 




If ^ = o be increased by equal angles, the correspond- 
ing values of r will be in geometrical progression. With 
any convenient radius, as FA^ describe a circle, and lay oft 
equal arcs Ab, bm, mc, etc. Draw the right lines PA^ Pb, 
Pm^ etc., and let PA be the initial side of ^. PA will then 
represent unity. Make ^ = Ab/PA, and determine the 
corresponding value of ^ = PB. PB/PA will be the ratio 
of the progression, and the distances PM^ PC, etc., from 
P to corresponding points of the curve are readily deter- 
mined. Since r = o requires 6^= — 00, the number of 
spires from A x.q P \^ unlimited. 



CHAPTER XX. 

APPLICATIONS TO SURFACES. 

196. u — F{x,y, z)= o . (i) or z =/{x,y) (2) 
is the general equation of any surface. 

Assuming the co-ordinate axes perpendicular to each 
other, and x and y as the independent variables, we have 

, du , . du du J , . 

du =. —- ax -\ — — dy -\- —- dz ^= o, . . . y-i) 

dx ' dy •" ' dz ' ^^' 

or 

'''=tJ'' + %'^^ -w 

for the general differential equation of the surface. 

I97« To find the equation of the tangent plane to any sur- 
face at a give?i point. 

Let z = (f>{x, y) be the equation of the surface and 
P(x\ y* , z') be the given point. 

Then ^. = |;y. + |;</v, (I) 

in which -r-, and -r-. are constant for a fixed point (§ 102). 
dx dy' 

The equations of the tangent line J^Q in the vertical 
plane PMN are, from the figure, 

x — x' _y ^ y' _z — z' 

. . . (2) 



dx dy dz 

348 



APPLICATIONS TO SURFACES. 



349 



Equations (i) and (2) are relations existing for any set 
of values of dx^ dy, and dz, corresponding to the point P 
and the corresponding tangent. 




Eliminating dx, dy, and dz^ we find the locus of all these 
tangents to be 

^-^'=^r{^--')-\-%(y-y)' . . (3) 



dx' 



If the equation of the surface be in form 
u = F{x, y, z) = o, 

then du = -—r dx -\- -^^ dy -{--—. dz = o, 

dx dy' -^ dz' 

in which dx, dy, and dz are the same as in (i). 

Combining this equation with (2), the equation of the 
tangent plane becomes 

9« / /\ t 9^^ / /\ I 9^^ / ,x / X 

-,{^-^)+^(>,-^)+-,(.-.')=o. (4) 



350 DIFFERENTIAL CALCULUS. 

(For tangent plane to surface of ^d order only, compare 

^ o • 1 . o 1-j ^ o \ A- Tr 9^' 9^ 9^ 

C. Smith s bond Geo., <5 1:52.) JNote. if — -, = t7 = "7~7 = o» 
^ ^ ^ dx' dy dz 

the plane is indeterminate. 

EXAMPLES. 

1. Find the equation of the tangent plane at the point 

(2, 3, 1^23) on the surface x^ -\- y^ -\- z^ ■=■ 36. 
"dz' /dx' = - x'/z' = - 2/ V^ 

a^V^y- -/A'- -3^^ 

Substituting in (3), we have 

^ _ V23 = (^ - 2)( - 2/ 1^23) + (y — 3)(— 3/ 1/^), 

or 2Jt: -j- 3^ + 4^230 = 36, for the required platie, 
or 

'du/dx' = 2x' = 4, du/dy' = 2j' = 6, du/dz' = 2z' = 2 V23 

Substituting in (4), we obtain same result. 

2. Find the equation of the tangent plane at {x\ y\ z') 
on the ellipsoid x^ /a 4-//^' + z" /c"" = i. 

Ans. xx'/a' -\-yy'/b'' + zz'/^ = i. 

3. Find the equation of the tangent plane at {x' yy\ 2') 
on the surface whose equation is 

mx^ -\-ny'^ -\-pz' -\- m''x-\-l=-o, 
Ans. 2mx'x + 2ny'y -\- 2pz'z + m"{x -\- x') + 2/ = o. 

4. Find the equation of the tangent plane to the ellipsoid 
whose equation is 

4jp' + 2/ + ^' = 10, at (i, — I, 2). 

•Ans. 2X —y -\-z=.^. 



APPLICATIONS TO SURFACES. 35 1 

5. Find the equation of the tangent phme to the ellipsoid 

^Vi6 +//9 + 2V4 = I, at (3, I, i/^/36). 

Ans. 3>:r/i6-h yh + ( ^47/36)^4 = i. 

6. Find the equation of the tangent plane at any point of 

2222 
the surface x^-\-y^ +2^ = ^^; and show that the sum of the 

squares of the intercepts on the axes made by a tangent 

plane is constant. 

198. The normal at any point of a surface. 

Let equation of surface be ;s = 0(-^>>')^ and point be 
{x\ y, 2;'). The normal passes through (jc', y\ z') and is 
perpendicular to the plane given by (3), § 197; hence its 
equations are 

X — x^ _ y —y' __ z — z' . . 

■ -dz'/dx' ~ dz'/dy' ~ ^=T ^'^ 

If equation of surface be « = -^{x^ y, z) = o, equations of 
normal are 



X y —y _ z — 



du/dx' du/dy' 'du/dz'' 



(2) 



(EXAMPLES. 



T. Deduce formulas for the direction cosines of the nor- 
mal given by (i); by (2). And find the cosines of the angles 
the tangent planes corresponding to (i) and (2) make with 
XY, XZ, and YZ. 

2. Find the tangent of the angle that the tangent plane 
to x" +y -^ ^"^ = Z^ at (2, 3, ^^23) makes with XY. 

199. To find the equations of the tangent line to a given 
surve at a given point. 



352 DIFFERENTIAL CALCULUS. 

Let equations of curve be 

F{x,y,z) = o, (a) 

(pipe, y,z) = o^ (b) 

and let {x\ y\ z') be the point of tangency. 

The required tangent line lies in the tangent plane to 
each of the surfaces {a) and {b) at point {x\ y\ z')\ hence 
its equations are 



9^/ ^\ 1 90 , /\ I 90/ ,x 



If the curve be given by two of its projecting cylinders as 

/(=c, ^)=o, {a') 

t(y,^) = o . (f) 

the equations of the tangent become 

f (.-/)+ |;(.-.')=o, 

which (§ 148) are the equations of lines in XZ and YZ 
respectively tangent to the curves {a') and {b') at the 
projections of (x , y\ z'). The problem is thus reduced to 
finding the equations of right lines tangent to two of the 



APPLICATIONS TO SURFACES. 



353 



projections of the given curve at the projections of the given 
point. 

Ex. Show that the curve whose equations are 



x^ +y = ^' aiid z = aC tan 



!>' 



makes a constant angle with the axis Z. 

200. To find the equation of the normal plane to a curve 
at a given point. 

Let the equations of the curve be 

F{x, y, z) = o and (p{x, y, z) = o, 

and (^', y, z') be the given point. 

The normal plane is perpendicular to the tangent to the 
curve at the given point. 

Let equation of normal plane be 

x{x- x') -^^{y -y ) + y{z - z') = o. 

If this plane be perpendicular to the tangent we have the 
conditions 

( .dF . dF ^ dF 
Eliminating A, //, and v^ the required equation is 



x — x, y~y , z — z 

dF dF dF 

dx" dy" dz' 

90 90 90 

dx" dy" dz' 



o. 



354 



D IFFEREN TIA L CALCUL US. 



20I. The numerical value of the expression 



dz/ Vdx' + d/ = tan QMT 

measures ^/le slope of the surface z = f {x,y), at the point 
Af, along any section, as 3fB, made bv a vertical plane 
through M, and whose trace on VX makes with X 
tan-^ dy/dx = (§ 102). <^ 




Writing tan QMT = tan i- = 



we have, (2), § 102, 



Vi + {dy/dxy 



tan s 



dz/dx -\- (dz/dy) (dy/dx) 
Vi + (dy/dxY 



Placing 

dy/dx = tan (p = m, dz/dx = /, and d^/dy = ^, 
we have 



tan i" = (/ + m^)/ J 1 -j- 7/^' (i) 



APPLICATIONS TO SURFACES. 355 

Application. — At tlie point (2, 3, ^^23) on the surface 

x" + / + ^^ = 36 . . . . (^) 

find the slope of the curve cut out by the plane 

y = 2X — \, Z — 0/0. ..... (/5) 

From (^), 'dz/dx = — x/z, and dz/dy = — y/z. 

Hence, j> = — 2/^23, and ^ — ~ 3/ ^^23. 

From (I?), dy/dx — 2 ^^ m. Therefore, 

tan J = [— 2/1/23 — 6/ 1/23]/ Vi -J- 4 " — 0.746 -{-. ^ 

Hence, 0.746 + is the required slope. 
202. At any point, as M, tan s varies with w. To deter- 
mine w, in order that the slope shall be a maximum, we 
place 

d tan s _ q — mp _ 
~"d^ ~ (i + vef'''' ~ ^' 

whence q — vip = o, or m = q/p. 

When tan s is positive, maximum values of tan s and the 
slope are the same ; but when tan s is negative, the slope is 
a maximum when tan i" is a minimum. 

Application. — Find the equation of the vertical plane 
which passes through the point (2, 3, I/23) on the surface, 
•^^ +y -|- s^ = T^d, and which cuts from the surface the 
line with the maximum slope. 

y — mx -\- b^ z = 0/0, is the general form of the required 
equation. 

At (2, 3, V^) p = dz/dx =-2/ V7s, 

and q = dz/dy = — 3/ 1/23. 



35^ DIFFERENTIAL CALCULUS. 

Hence, m = 3/2. The trace on XY must pass through 
(2, 3). Hence, we have 



3 + /^, or /^ — o, and y = Z^l'^i ^ ~ °/° 



is the required plane. The maximum slope is approxi- 
mately .751. 

When / -f" mq = o, or m = — p/q^ tan j- = o, and 
the slope is a minimum, since numerical values only of 
tan s are considered. 

In the above application m = — p/q = — 2/3. Hence, 
3 = — 4/3 4- /^, giving <^ = 13/3, and ^ = — 2^3 + 13/3, 
z = 0/0, is the plane which cuts out the curve whose 
tangent at M is parallel to XV. 

The intersection of the surface by the horizontal plane 
through the given-point is a horizontal line,and J^{x^y,c)=^o, 
z^c^ ars its equations. 



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Essentials of Volumetric Analysis , i2mo, 

Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco. 

Handbook for Sugar^anufacturers and their Chemists. . i6mo, morocco, 
Stockbridge's Rocks and Soils 8vo, 

* Tillman's Elementary Lessons in Heat 8vo, 

♦ Descriptive General Chemistry 8vo, 

Treadwell's Qualitative Analysis. (Hall.) 8vo, 

Quantitative Analysis. (Hall.) Svo, 

Turneaure and Russell's Public Water-supplies Svo, 

Van Deventer's Physical Chemistry for Beginners. (Boltwood.) i2mo, 

* Walke's Lectures on Explosives Svo, 

Wassermann's Immune Sera: Haemolysins, Cytotoxins, and Precipitins. (Bol- 

duan.) i2mo, 

Wells's Laboratory Guide in Qualitative Chemical Analysis Svo, 

Short Course in Inorganic Qualitative Chemical Analysis for Engineering 

Students i2mo, 

Whipple's Microscopy of Drinking-water Svo, 

Wiechmann's Sugar Analysis Small Svo. 

Wilson's Cyanide Processes. i2mo, 

Chlorination Process i2mo. 

Wulling's Elementary Course in Inorganic Pharmaceutical and Medical Chem- 
istry. i2mo, 2 .00 

CIVIL ENGINEERING. 

BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING 
RAILWAY ENGINEERING. 

Baker's Engineers' Surveying Instruments i2mo, 3 00 

Bixby's Graphical Computing Table Paper 19^X24! inches. 25 

** Burr's Ancient and Modern Engineering and the Isthmian Canal. (Postage, 

27 cents additional.) Svo, net, 3 50 

Comstock's Field Astronomy for Engineers Svo, 2 50 

Davis's Elevation and Stadia Tables Svo, i 00 

Elliott's Engineering for Land Drainage i2mo, i 50 

Practical Farm Drainage i2mo, z oc 

Folwell's Sewerage. (Designing and Maintenance.) Svo, 3 00 

Freitag's Architectural Engineering. 2d Edition, Rewritten Svo, 3 so 

French and Ives's Stereotomy Svo, 3 50 

Goodhue's Municipal Improvements i2mo, t 75 

Goodrich's Economic Disposal of Towns' Refuse Svo, 3 50 

Gore's Elements of Geodesy Svo, 2 50 

Hayford's Text-book of Geodetic Astronomy Svo, 3 01: 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Howe's Retaining Walls for Earth i2mo, i 25 

Johnson's Theory and Practice of Surveying Small Svo, 4 00 

Statics by Algebraic and Graphic Methods ,♦...., Svo, a 00 

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Kiersted's Sewage Disposal i2mo, i as 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 2 00 

Mahan's Treatise on Civil Engineering. (1873O (Wood.) 8vo, 5 00 

♦ Descriptive Geometry 8vo, i 50 

Merriman's Elements of Precise Surveying and Geodesy Svo, 2 50 

Elements of Sanitary Engineering Svo, 2 00 

Merriman and Brooks's Handbook for Surveyors i6m0( morocco, 2 00 

Nugent's Plane Surveying. Svo, 3 50 

Ogden's Sewer Design i2mo, 2 00 

Patton's Treatise on Civil Engineering Svo half leather, 7 50 

Reed's Topographical Drawing and Sketching 4to, 5 00 

Rideal's Sewage and the Bacterial Purification of Sewage Svo, 3 51 

Siebert and Biggin's Modem Stone-cutting and Masonry Svo, i 5* 

Smith's Manual of Topographical Drawing, (McMillan.) Svo, 2 50 

Sondericker's Graphic Statics, wun Applications to Trusses. Beams, and 

Arches Svo, 2 00 

* Traxitwine's Civil Engineer's Pocket-book i6mo, morocco, 5 00 

Wait's Engineering and Architectural Jurisprudence Svo, 6 00 

Sheep, 6 50 
Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture. Svo, 5 00 

Sheep, 5 50 

Law of Contracts Svo, 3 00 

Warren's Stereotomy — Problems in Stone-cutting Svo, 2 50 

Webb's Problems in the Use and Adjustment of Engineering Instruments. 

i6mo, morocco, i 25 

• Wheeler's Elementary Course of Civil Engineering Svo, 4 00 

Wilson's Topographic Surveying , , Svo, 3 50 

BRIDGES AND ROOFS. 

Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .Svo, 2 00 

* Thames River Bridge 4to, paper, 5 00 

Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and 

Suspension Bridges Svo, 3 50 

Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 00 

Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 

Fowler's Coffer-dam Process for Piers Svo, 2 50 

Greene's Roof Trusses Svo, i 25 

Bridge Trusses « Svo, 2 50 

Arches in Wood, Iron, and Stone Svo, 2 50 

Howe's Treatise on Arches Svo, 4 00 

Design of Simple Roof-trusses in Wood and Steel Svo, 2 00 

J«hnson, Bryan, and Tumeaure's Theory and Practice in the Designing of 

Modern Framed Structures Small 4to, 10 00 

Merriman and Jacoby's Text-book on Roofs and Bridges: 

Part I. — Stresses in Simple Trusses Svo, 2 50 

Part n. — Graphic Statics Svo, 2 50 

Part III. — Bridge Design. 4th Edition, Rewritten Svo, 2 50 

Part IV. — Higher Structures Svo, 2 50 

Morison's Memphis Bridge 4to, 10 00 

Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo, morocco, 3 00 

Specifications for Steel Bridges i2mo, i 25 

Wood's Treatise on the Theory of the Construction of Bridges and Roofs. Svo, 2 00 
Wright's Designing of Draw-spans: 

Part I. — Plate-girder Draws Svo, 2 50 

Part II. — Riveted- truss and Pin-connected Long-span Draws Svo, 2 50 

Two parts in one volume Svo, 3 50 

6 



^ HYDRAULICS. 

Bazin*s Experiments upon the Contraction of the Liquid Vein Issuing from an 

Orifice. (Trautwine.) 8vo, 2 00 

Bovey's Treatise on Hydraulics 8vo, 5 00 

Church's Mechanics of Engineering 8vo, 6 00 

Diagrams of Mean Velocity of Water in Open Channels paper, i 50 

Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 

Flather's Dynamometers, and the Measurement of Power i2mo, 3 00 

Folwell's Water-supply Engineering , Svo, 4 00 

Frizell's Water-power Svo, 5 00 

Fuertes's Water and Public Health , i2mo, i 50 

Water-filtration Works i2mo, 2 50 

Ganguillet and Kutter's General Formula for the Uniform Flow of Water in 

Rivers and Other Channels. (Hering and Trautwine.) Svo, 4 00 

Hazen's Filtration of Public Water-supply Svo, 3 00 

Hazlehurst's Towers and Tanks for Water- works Svo, 2 50 

Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal 

Conduits Svo, 2 00 

Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten Svo, 4 00 

Merriman's Treatise on Hydraulics. 9th Edition, Rewritten Svo, 5 00 

* Michie's Elements of Analytical Mechanics Svo, 4 00 

Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- 
supply : Large Svo, 5 00 

♦* Thomas and Watt's Improvement of Riyers. (Post., 44 c. additional), 4to, 6 00 

Turneaure and Russell's Public Water-supplies Svo, 5 00 

Wegmann's Desien and Construction of Dams 4to, 5 00 

Water-supply of the City of New York from 1658 to 1895 4to, 10 00 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.) Svo, 5 00 

Wilson's Manual of Irrigation Engineering Small Svo, 4 00 

Wolff's Windmill as a Prime Mover Svo, 3 00 

Wood's Turbines Svo, 3 50 

Elements of Analytical Mechanics Svo, 3 00 

MATERIALS OF ENGINEERING. 

Baker's Treatise on Masonry Construction 8vo, 5 00 

Roads and Pavements Svo, s 00 

Black's United States Public Works Oblong 4to, 5 00 

Bovey's Strength of Materials and Theory of Structures Svo, 7 50 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edi- 
tion, Rewritten Svo, 7 50 

Byrne's Highway Construction Svo, 5 00 

Inspection of the Materials and Workmanship Employed Jn Construction. 

i6mo, 3 00 

Church's Mechanics of Engineering Svo, 6 00 

Du Bois's Mechanics of Engineering. Vol. I Small 4to, 7 50 

Johnson's Materials of Construction Large Svo, 6 00 

Keep's Cast Iron Svo, 2 50 

Lanza's Applied Mechanics Svo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) 2 vols Svo, 7 50 

Merrill's Stones for Building and Decoration Svo, 5 00 

Merriman's Text-book on the Mechanics of Materials Svo, 4 00 

Strength of Materials i2mo, i 00 

Metcalf's Steel. A Manual for Steel-users i2mo, 2 00 

Patton's Practical Treatise on Foundations Svo, 5 00 

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Rockwell's Roads and Pavements in France i2mo, 

Smith's Materials of Machines i2mo, 

Snow's Principal Species of Wood 8vo, 

Spalding's HydrauUc Cement i2mo. 

Text-book on Roads and Pavements i2mo, 

Thurston's Materials of Engineering. 3 Parts 8vo, 

Part I. — Non-metallic Materials of Engineering and Metallurgy 8vo, 

Part II.— Iron and Steel 8vo, 

Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo, 

Thurston's Text-book of the Materials of Construction 8vo, 

TiUson's Street Pavements and Paving Materials 8vo, 

Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor.. 

Specifications for Steel Bridges i2mo. 

Wood's Treatise on the Resistance of Materials, and an Appendix on the Pres- 
ervation of Timber 8vo, 

Elements of Analytical Mechanics 8vo, 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .Svo, 

RAILWAY ENGINEERING. 

Andrews's Handbook for Street Railway Engineers. 3X5 inches, morocco, i 25 

Berg's Buildings and Structures of American Railroads 4to, s 00 

Brooks's Handbook of Street Railroad Location i6mo. morocco, i 50 

Butts's Civil Engineer's Field-book i6mo, morocco, 2 50 

Crandall's Transition Curve i6mo, morocco, i 50 

Railway and Other EarthworkvTables 8vo, i 50 

Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 
Dredge's History of the Pennsylvania Railroad : (1879) Paper, 5 00 

• Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., 25 00 

Fisher's Table of Cubic Yards Cardboard, 25 

Godwin's Railroad Engineers' Field-book and Explorers' Guide i6mo, mor., 2 50 

Howard's Transition Curve Field-book i6mo, morocco, i so 

Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments 8vo, I 00 

Molitor and Beard's Manual for Resident Engineers i6mo, i 00 

Nagle's Field Manual for Railroad Engineers i6mo, morocco. 3 00 

Philbrick's Field Manual for Engineers i6mo, morocco, 3 00 

Searles's Field Engineering i6mo, morocco, 3 00 

Railroad SpiraL i6mo, morocco, i 50 

Taylor's Prismoidal Formulae and Earthwork Svo, 1 50 

• Trautwine's Method of Calculating the Cubic Contents of Excavations and 

Embankments by the Aid of Diagrams Svo, 2 00 

The Field Practice of [Laying Out Circular Curves for Railroads. 

1 2mo, morocco, 2 50 

Cross-section Sheet Paper, 25 

Webb's Railroad Construction. 2d Edition, Rewritten i6mo. morocco, s 00 

Wellington's Economic Theory of the Location of Railways Small Svo, 5 00 

DRAWING. 

Barr's Kinematics of Machinery Svo, 2 50 

• Bartlett's Mechanical Drawing Svo, 3 00 

• •• ' " Abridged Ed Svo, i 50 

Coolidge's Manual of Drawing 8vo, paper, i 00 

Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- 
neers. {In press.) 

Durley's Kinematics of Machines — . , Svo, 4 oe 

8 



Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 oo 

Jamison's Elements of Mechanical Drawing. {In press.) 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, i 50 

Part II. — Form, Strength, and Proportions of Parts 8vo, 3 00 

MacCord's Elements of Descriptive Geometrj . , 8vo, 3 00 

Kinematics; or. Practical Mechanism , , , 8vo, 5 00 

Mechanical Drawing < . . 4to, 4 00 

Velocity Diagrams 8vo, i so 

• Mahan's Descriptive Geometry and Stone-cutting 8vo, i 50 

Industrial Drawing. (Thompson.) 8vo, 3 50 

Reed's Topographical Drawing and Sketching 4to, 5 00 

Reid's Course in Mechanical Drawing 8vo, 2 00 

Text-book of Mechanical Drawing and Elementary Machine Design . . 8vo, 3 o^ 

Robinson's Principles of Mechanism 8vo, 3 00 

Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 

Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. . i2mo, i 00 

Drafting Instruments and Operations i2mo, i 25 

Manual of Elementary Projection Drawing i2mo, i 50 

Manual of Elementary Problems in the Linear Perspective of Form and 

Shadow izmo, i 00 

Plane Problems in Elementary Geometry i2mo, i 25 

Primary Geometry ^ . i2mo, 73 

Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50 

General Problems of Shades and Shadows 8to, 3 00 

Elements of Machine Construction and Drawing 8vo, 7 50 

Problems. Theorems, and Examples in Descriptive Geometrv 8vo, 2 50 

Weisbach's Kinematics and the Power of Transmission. (Hermann and 

Klein.) 8vo, 5 00 

Whelpley's Practical Instruction in the Art of Letter Engraving i2mo, 2 00 

Wilson's Topographic Surveying 8vo, 3 50 

Free-hand Perspective 8vo, 2 50 

Free-hand Lettering 8vo, x 00 

Wooif's Elementary Course in Descriptive Geometry Large 8vo, 3 00 

"ELECTRICITY AND PHYSICS. 

Anthony and Brackett's Text-book of Physics. (Magie.) Small 8vo, 3 00 

Anthony's Lecture-notes on the Theory of Electrical Measurements i2mo, i 00 

Benjamin's History of Electricity 8vo, 3 00 

Voltaic Cell. 8vo, 3 00 

Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 3 00 

Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 00 

Dawson's "Encineering" and Electric Traction Pocket-book. .i6mo, morocco, 5 00 
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende.) i2mo,"2 50 

Duhem's Thermodynamics and Chemistry. (Burgess.) Svo, 4 00 

Flather's Dvnamometers, and the Measurement of Power i2mo, 3 00 

Gilbert's De Magnete. (Mottelay.) 8vo, 2 50 

Hanchett's Alternating Currents Explained i2mo, i 00 

Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Holman's Precision of Measurements 8vo, 2 00 

Telescopic Mirror-scale Method, Adjustments, and Tests Large Svo, 75 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess. )i2mo, 3 00 

Lob's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) i2mo, i 00 

• Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and U. Svo, each, 6 00 

• Michie. Elements of Wave Motion Relating to Sotmd and Light Svo, 4 00 



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Niaudet's Elementary Treatise on Electric Batteries. (FisHoack. )...... izmo, 2 50 

• Rosenberg's Electrical Engineering, (Haldane Gee — Kinzbrunner.). . . .8vo, i 50 

Ryan, Korris, and Hoxie's Electrical Machinery. VoL 1 8vo, 2 50 

Thurston's Stationary Steam-engines 8vo, 2 50 

* Tillman's Elementary Lessons in Heat 8vo, i 50 

Tory and Pitcher's Manual of Laboratory Physics Small 8vo, 2 00 

Ulke's Modern Electrolytic Copper Refining 8vo, 3 00 



LAW. 

* Davis's Elements of Law 8vo, 

* Treatise on the Military Law of United States 8vo, 

* Sheep, 

Manual for Courts-martial i6mo, morocco, 

Wait's Engineering and Architectural Jurisprudence 8vo, 

Sheep, 
Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo, 

Sheep, 

Law of Contracts 8vo, 

Winthrop's Abridgment of Military Law i2mo, 

MANUFACTURES. 

Bernadou's Smokeless Powder — Nitro-cellulose and Theory of the Cellulose 

Molecule i2mo, 2 so 

Bolland's Iron Founder i2mo, 2 50 

*• The Iron Founder," Supplement i2mo, 2 50 

Encyclopedia of Founding and Dictionary of Foundry Terms Used in the 

Practice of Moulding i2mo, 3 00 

Eissler's Modem High Explosives 8vo, 4 00 

Effront's Enzymes and their Applications, (Prescott,) Svo, 3 00 

Fitzgerald's Boston Machinist iSmo, i 00 

Ford's Boiler Making for Boiler Makers i8mo, i 00 

Hopkins's Oil-chemists' Handbook Svo, 3 00 

Keep's Cast Iron Svo, 2 50 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 

Metcalf's SteeL A Manual for Steel-users i2mo, 2 00 

Metcalfe's Cost of Manufactures — And the Administration of Workshops, 

Public and Private Svo, 5 00 

Meyer's Modern Locomotive Construction 4to, 10 00 

Morse's Calculations used in Cane-sugar Factories. i6mo, morocco, i 50 

* Reisig's Guide to Piece-dyeing Svo, 25 00 

Smith's Press-working of Metals Svo, 3 00 

Spalding's Hydraulic Cement i2mo, 2 00 

Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 

HandbooK lor sugar Manutaciurers and their Chemists.. . i6mo, morocco, 2 00 
Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- 
tion Svo, s 00 

* Walke's Lectures on Explosives Svo, 4 00 

West's American Foundry Practice i2mo, 2 50 

Moulder's Text-book , i2mo, 2 50 

Wiechmann's Sugar Analysis Small 8vo, 2 50 

Wolff's Windmill as a Prime Mover Svo, 3 00 

Woodbury's Fire Protection of Mills Svo, 2 50 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .. Svo, 4 00 

10 



MATHEMATICS. 

Baker's Elliptic Functions 8vo, 

* Bass's Elements of Differential Calculus i2mo, 

Briggs's Elements of Plane Analytic Geometry i2mo, 

Compton's Manual of Logarithmic Computations i2mo, 

Davis's Introduction to the Logic of Algebra 8vo, 

* Dickson's College Algebra Large i2mo, 

* Answers to Dickson's College Algebra 8vo, paper, 

* Introduction to the Theory of Algebraic Equations Large i2mo, 

Halsted's Elements of Geometry 8vo, 

Elementary Synthetic Geometry Svo, 

Rational Geometry i2mo, 

* Johnson's Three-place Logarithmic Tables : Vest-pocket size paper, 

loo copies for 

* Mounted on heavy cardboard, 8 X lo inches, 

10 copies for 

Elementary Treatise on the Integral Calculus Small Svo, 

Curve Tracing in Cartesian Co-ordinates i2mo. 

Treatise on Ordinary and Partial Differential Equations Small Svo, 

Theory of Errors and the Method of Least Squares i2mo, 

* Theoretical Mechanics i2mo, 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 

* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other 

Tables Svo, 

Trigonometry and Tables published separately Each, 

* Ludlow's Logarithmic and Trigonometric Tables Svo, 

Maurer's Technical Mechanics Svo, 

Merriman and Woodward's Higher Mathematics Svo, 

Merriman's Method of Least Squares Svo, 

Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., Svo, 

Differential and Integral Calculus. 2 vols, in one Small Svo, 

Sabin's Industrial and Artistic Technology of Paints and Varnish. {In press.) 
Wood's Elements of Co-ordinate Geometry Svo, 

Trigonometry: Analytical, Plane, and Spherical i2mo, 

MECHANICAL ENGINEERING. 
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 

Baldwin's Steam Heating for Buildings i2mo, 

Barr's Kinematics of Machinery . . , . ; Svo, 

* Bartlett's Mechanical Drawing Svo, 

* " " " Abridged Ed Svo, 

Benjamin's Wrinkles and Recipes i2mo. 

Carpenter's Experimental Engineering Svo, 

Heating and Ventilating Buildings Svo, 

Gary's Smoke Suppression in Plants using Bituminous CoaL (In prep- 
aration.) 

Clerk's Gas and Oil Engine Small Svo, 

Coolidge's Manual of Drawing Svo, paper, 

CooUdge and Freeman's Elements of General Drafting for Mechanical En- 
gineers. {In press.) 

Cromwell's Treatise on Toothed Gearing i2mo. 

Treatise on Belts and Pulleys i2mo, 

Durley's Kinematics of Machines Svo, 

Flather's Dynamometers and the Measurement of Power i2mo. 

Rope Driving , , i2mo, 

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Gill's Gas and Fuel Analysis for Engineers i2mo, i 25 

Hall's Car Lubrication i2mo, i 00 

Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Button's The Gas Engine 8vo, s ou 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, i 50 

Part II. — Form, Strength, and Proportions of Parts 8vo, 3 00 

Kent's Mechanical Engineer's Pocket-book i6mo, morocco, 5 00 

Kerr's Power and Power Transmission Svo, 2 00 

MacCord's Kinematics; or, Practical Mechanism 8vo, 5 00 

Mechanical Drawing 4to, 4 00 

Velocity Diagrams 8vo, i 50 

Mahan's Industrial Drawing. (Thompson.) 8vo, 3 50 

Poole's Calorific Power of Fuels ,^vo, 3 00 

Reid's Course in Mechanical Drawing 8vo, 2 00 

Text-book of Mechanical Drawing and Elementary Machine Design. ,8vo, 3 00 

Richards's Compressed Air i2mo, i 50 

Robinson's Principles of Mechanism 8vo, 3 00 

Smith's Press-working of Metals ,8vo, 3 00 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 

Work 8vo, 3 00 

Animal as a Machine and Prime Motor, and the Laws of Energetics. 1 2mo, i 00 

Warren's Elements of Machine Construction and Drawing 870, 7 50 

Weisbach's Kinematics and the Power of Transmission. Herrmann — 

Klein.) Svo, 5 00 

Machinery of Transmission and Governors. (Herrmann — Klein.). . Svo, 500 

Hydraulics and Hydraulic Motors. (Du Bois.) Svo, 5 00 

Wolff's Windmill as a Prime Mover Svo, 3 00 

Wood's Turbines Svo, 2 50 

MATERIALS OF ENGINEERING. 

Bovey's Strength of Materials and Theory of Structures Svo, 7 50 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition, 

Reset Svo, 7 50 

Church's Mechanics of Engineering Svo, 6 00 

Johnson's Materials of Construction Large Svo, 6 00 

Keep's Cast Iron Svo, 2 50 

Lanza's Applied Mechanics Svo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) Svo, 7 50 

Merriman's Text-book on the Mechanics of Materials Svo, 4 00 

Strength of Mater>als i2mo, i 00 

Metcalf's SteeL A Manual for Steel-users i2mo, 2 00 

Smith's Materials of Machines i2mo i 00 

Thurston's Materials of Engineering 3 vols., Svo, 8 00 

Part II.— Iron and Steel Svo, 3 50 

Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents Svo 2 50 

Text-book of the Materials of Construction Svo, 5 00 

Wood's Treatise on the Resistance of Materials and an Appendix on the 

Preservation of Timber Svo, 2 00 

Elements of Analytical Mechanics Svo, 3 00 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel, . .Svo, 4 00 

STEAM-ENGINES AND BOILERS. 

Carnot's Reflections on the Motive Power of Heat. (Thtirston.) i2mo, l 50 

Dawson's "Engineering" and Electric Traction Pocket-book. .i6mo, mor., 5 00 

Ford's Boiler Making for Boiler Makers iSmo, i 00 

12 



Goss's LocomotiTe Sparks 8vo, 2 00 

Hemenway's Indicator Practice and Steam-engine Economy i2mo, a 00 

Button's Mechanical Engineering of Power Plants 8vo, 5 00 

Heat and Heat-engines 8vo, 5 00 

Kent's Steam-boiler Economy Svo, 4 00 

Kneass's Practice and Theory of the Injector Svo i 50 

MacCord's Slide-valves Svo, 2 00 

Meyer's Modem Locomotive Construction 4to» 10 00 

Peabody's Manual of the Steam-engine Indicator i2mo, i 50 

Tables of the Properties of Saturated Steam and Other Vapors Svo, i 00 

Thermod3mamics of the Steam-engine and Other Heat-engines Svo, 5 oo 

Valve-gears for Steam-engines Svo, 2 50 

Peabody and Miller's Steam-boilers Svo, 4 00 

Pray's Twenty Years with the Indicator Large Svo, 2 50 

Pupln's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 

(Osterberg.) i2mo, i 25 

Reagan's Locomotives : Simple, Compound, and Electric i2mo, 2 50 

Rontgen's Principles of Thermodsmamics. (Du Bois.) Svo, 5 00 

Sinclair's Locomotive Engine Running and Management i2mo, 2 00 

Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 

Snow's Steam-boiler Practice Svo, 3 00 

Spangler's Valve-gears Svo, 2 5© 

Notes on Thermodynamics i i2mo, i 00 

Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00 

Thurston's Handy Tables Svo, i 50 

Manual of the Steam-engine 2 vols. . Svo, 10 00 

Part I. — History, Structuce, and Theory Svo, 6 00 

Part n. — Design, Construction, and Operation Svo, 6 00 

Handbook of Engine and Boiler Trials, and the Use of the Indicator and 

the Prony Brake Svo 5 oe 

Stationary Steam-engines Svo, 2 50 

Steam-boiler Explosions in Theory and in Practice i2mo i 50 

Manual of Steam-boilers, Their Designs, Construction, and Operation .Svo, 5 00 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 o« 

Whitham's Steam-engine Design Svo, 5 00 

Wilson's Treatise on Steam-boilers. (Flather.) i6mo, 2 50 

Wood's Thermodynamics Heat Motors, and Refrigerating Machines. . . .Svo, 4 00 



MECHANICS AND MACHINERY. 



Barr's Kinematics of Machinery Svo, 

Bovey's Strength of Materials and Theory of Structures Svo, 

Chase's The Art of Pattern-making i2mo, 

Chordal. — Extracts from Letters i2mo, 

Church's Mechanics of Engineering Svo, 

Notes and Examples in Mechanics Svo, 

Compton's First Lessons in Metal- working lamo, 

Compton and De Groodt's The Speed Lathe i2mo, 

Cromwell's Treatise on Toothed Gearing i2mo, 

Treatise on Belts and Pulleys i2mo, 

Dana's Text-book of Elementary Mechanics for the Use of Colleges and 

Schools i2mo. 

Dingey's Machinery Pattern Making i2mo. 

Dredge's Record of the Transportation Exhibits Building of the World's 

Columbian Exposition of iSgs 4to, half morocco, 5 00 

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Du Bois's Elementary Principles of Mechanics : 

Vol. I. — Kinematics 8vo, 

Vol. II.— Statics .' 8vo, 

Vol. III.— Kinetics 8vo, 

Mechanics of Engineering. Vol. I Small 4to, 

Vol. II Small 4to, 

Durley's Kinematics of Machines 8vo, 

Fitzgerald's Boston Machinist i6mo, 

Flather's Dynamometers, and the Measurement of Power i2mo. 

Rope Driving i2mo, 

Goss's Locomotive Sparks 8vo 

Hall's Car Lubrication i2mo. 

Holly's Art of Saw Filing i8mo , 

* Johnson's Theoretical Mechanics i2mo. 

Statics by Graphic and Algebraic Methods 8vo, 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 

Part n. — Form, Strength, and Proportions of Parts 8vo, 

Ketr's Power and Power Transmission 8vo, 

Lanza's Applied Mechanics 8vo, 

MacCord's Kinematics; or. Practical Mechanism 8vo, 

Velocity Diagrams 8vo, 

Maurer's Technical Mechanics 8vo, 

Merriman's Text-book on the Mechanics of Materials 8vo, 

• Michie's Elements of Anal3^ical Mechanics 8vo, 

Reagan's Locomotives: Simple, Compound, and Electric i2mo, 

Reid's Course in Mechanical Drawing 8vo, 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 

Richards's Compressed Air i2mo, 

Robinson's Principles of Mechanism 8vo, 

Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 

Sinclair's Locomotive-engine Running and Management i2mo, 

Smith's Press-working of Metals 8vo, 

Materials of Machines i2mo, 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 
Work 8vo, 

Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, 

Warren's Elements of Machine Construction and Drawing 8vo, 

Weisbach's Kinematics and the Power of Transmission. (Herrmann — 
Klein.) 8vo, 

Machinery of Transmission and Governors. (Herrmann — Klein. ).8vo. 
Wood's Elements of Analytical Mechanics 8vo, 

Principles of Elementary Mechanics i2mo. 

Turbines 8vo, 

The World's Columbian Exposition of 1893 4to, 

METALLURGY. 

Egleston's Metallurgy of Silver, Gold, and Mercury: 

Vol. I.— Silver 8vo, 7 So 

Vol. II.— Gold and Mercury 8vo, 7 50 

♦♦ Iles's Lead-smelting. (Postage 9 cents additional.) i2mo, 2 50 

Keep's Cast Iron 8vo, 2 50 

Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) . i2mo, 3 00 

Metcalf's SteeL A Manual for Steel-users i2mo, 2 00 

Smith's Materials of Machines i2mo, i 00 

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Thurston's Materials of Engineering, In Three Parts 8vo, 8 00 

Part II. — Iron and Steel Svo, 3 5o 

Part III. — A Treatise on Brasses. Bronzes, and Other Alloys and their 

Constituents Svo, 2 50 

Ulke's Modem Electrolytic Copper Refining Svo, 3 00 

MINERALOGY. 

Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 

Boyd's Resources of Southwest Virginia Svo, 

Map of Southwest Virginia Pocket-book form. 

Brush's Manual of Determinative Mineralogy. (Penfield.) Svo, 

Chester's Catalogue of Minerals Svo, paper. 

Cloth, 

Dictionary of the Names of Minerals „ Svo, 

Dana's System of Mineralogy Large Svo, half leather. 

First Appendix to Dana's New "System of Mineralogy.". .. .Large Svo, 

Text-book of Mineralogy Svo, 

Minerals and How to Study Them i2mo. 

Catalogue of American Localities of Minerals Large Svo, 

Manual of Mineralogy and Petrography i2mo, 

Eakle's Mineral Tables Svo, 

Egleston's Catalogue of Minerals and Synonyms Svo, 

Hussak's The Determination of Rock-forming Minerals. (Smith.) Small Svo, 
Merrill's Non-metallic Minerals: Their Occurrence and Uses Svo, 

* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 

Svo, paper, o 50 
Rosenbusch's Microscopical Physiography of the Rock-making Minerals. 

(Iddmgs.) Svo, 5 00 

• Tillman's Text-book of Important Minerals and Docks Svo, 2 00 

WiUiams's Manual of Lithology Svo, 3 00 

MINING. 

Beard's Ventilation of Mines i2mo, 2 50 

Boyd's Resources of Southwest Virginia Svo, 3 00 

Map of Southwest Virginia Pocket-book form, 2 00 

♦ Drinker's Tunneling, Explosive Compounds, and Rock Drills. 

4to, half morocco, 2S 00 

Eissler's Modem High Explosives Svo, 

Fowler's Sewage Works Analyses i2mo, 

Goodyear 's Coal-mines of the Western Coast of the United States i2mo, 

Ihlseng's Manual of Mining Svo, 

** Iles's Lead-smelting. (Postage 9c. additional) i2mo, 

Kunhardt's Practice of Ore Dressing in Europe Svo, 

O'Driscoll's Notes on the Treatment of Gold Ores Svo, 

* Walke's Lectures on Explosives Svo, 

Wilson's Cyanide Processes i2mo, 

Chlorination Process i2mo. 

Hydraulic and Placer Mining i2mo. 

Treatise on Practical and Theoretical Mine Ventilation i2mo 

SANITARY SCIENCE. 

Copeland's Manual of Bacteriology. (In preparation.) 

Folwell's Sewerage. (Designing, Construction and Maintenance.) Svo, 3 00 

Water-supply Engineering Svo, 4 00 

Fuertes's Water and Public Health i2mo, i 50 

Water-filtration Works i2mo, 2 50 

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Gerhard's Guide to Sanitary House-inspection i6mo, i 

Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 3 

Hazen's Filtration of Public Water-suppUes 8vo, 3 

Kiersted's Sewage Disposal i2mo, i 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 
Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten 8vo, 4 

Examination of Water. (Chemical and BacteriologicaL) i2mo, i 

Merriman's Elements of Sanitary Engineering , .- , 8vo, 2 

Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary 

Standpoint.) (1883.) Svo. 2 

Ogden's Sewer Design i2mo, 2 

Prescott and Winslow's Elements of Water Bacteriology, with Special Reference 

to Sanitary Water Analysis i2mo5 i 

* Price's Handbook on Sanitation i2mo, i 

Richards's Cost of Food. A Study in Dietaries i2mo, i 

Cost of Living as Modified by Sanitary Science i2mo, i 

Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
point 8vo, 2 

♦ Richards and Williams's The Dietary Computer 8vo, 1 

Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 

Turneaure and Russell's Public Water-supplies 8vo, s 

Whipple's Microscopy of Drinking-water 8vo, 3 

Woodhull's Notes and Military Hygiene i6mo, i 



MISCELLANEOUS. 

Barker's Deep-sea Soundings 8vo, 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

International Congress of Geologists Large 8vo 

Ferrel's Popular Treatise on the Winds 8vo 

Haines's American Railway Management I2mo^ 

Mott's Composition.'Digestibility . and Nutritive Value of Food. Mounted chart. 

Fallacy of the Present Theory of Sound i6mo 

Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, 

Rotherham's Emphasized New Testament Large 8vo, 

Steel's Treatise on the Diseases of the Dog 8vo, 

Totten's Important Question in Metrology 8vo 

The World's Columbian Exposition ot 1893 4to, 

Worcester and Atkinson. Small Hospitals, Establishment and Maintenance, 
and Suggestions for Hospital Architecture, with Plans for a Small 
Hospital i2mo. 



HEBREW AND CHALDEE TEXT-BOOKS. 

Green's Grammar of the Hebrew Language 8vo, 3 

Elementary Hebrew Grammar i2mo, i 

Hebrew Chrestomathy 8vo, 2 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(Tregelles.) Small 4to, half morocco, 5 

Letteris'a Hebrew Bible 8vo, 2 

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